Vol. 1. STRAIN HARDENING and SOFTENING of METALS

Vol. 1.

STRAIN HARDENING AND SOFTENING OF METALS PRODUCED BY CYCLES OF PLASTIC DEFORMATION

by P. W. J. OLDROYD B.Sc.(Eng.), A.C.G.I., D.I.C., A.M.I.Mech.E..

Thesis submitted for the degree of Doctor of Philosophy in the

Facelty of Engineering of the: University of London

Mechanical Engineering Department January 1968 City and Guilds College Imperial College of Science and Technology ... 7)-

Cyclic tansion-compression tests between fixed limits of plastic strain, some tests between fixed limits of stress and certain other tests were made on the- following materials: 014'i-IC co per and aluminium in the annealed and cold-worked states.; En25 steel in the annealed and hardened and tempered states; and fully soft austenitic stainless steel. Copper arid aluminium subjected to cyclic plastic deformation between strain limits harden or soften according to their initial state: and the strain amplitude.; they behave in a similar manner when stress limits are used. There appears to be a settled cyclic state for each strain (or stress ) amplitude. In stress cycling a progressive change of the mean strain (cyclic creep) is observed. This is very sensitive to changes in the mean stress and it is suggested that it will always develop, whatever the value of the mean stress, because of geometrical instability. Annealed or tempered En25 steel has a yield step in its stress-strain curve for the -first loading but if the yield stress is exceeded. the step is not present in the curves for immediate re-loading in the same, or opposite, direction. If the strain is not sufficiently great for this then cyclic deformation causes a gradual removal of the conditions responsible for the stop and and this change reveals itself in a softening of the material. If the amplitude of plastic strain is small considerabl softening can occur. The annealed steel does not significantly soften when cycled at strain amplitudes which are sufficiently large to eliminate the yield step at the first loading. The hardened steel softens significantly at all strain amplitudes near to or above the yield step strain. Soft austenitic stainless steel at first behaves in a similar manner to copper and aluminium but after a number of cycles - a number which depends on the strain amplitude - a sudden transition to rapid hardening occurs. The elimination of • asymmetry between the tension and compression properties (the Bauschinger effect) by the use of decreasing cycles of plastic deformation suggests that hardness - as measured by these. properties - has both intrinsic and extrinsic components. Simple analogue models for cyclic behaviour show only phenomena as:ic,ciated with the latter effect. A Method of predicting the cyclic behaviour of some metals from tne results obtained from simple reversed loading tests is proposed. -3-

ACKNOWLEDGMENTS'

The author has pleasure in recording his indebtedness to Dr. P. P. Benham

and Dr. D. J. Burns: to the former for having_introduced him to this field

of research and encouraged him to undertake the present work; and to the

latter for his guidance in the development of the investigation and - in

particular - for his eneonragement in thetaek'of interpreting anddeser

ibing the results of thereseereh.

The laboratory and workshop staff of the Department of Mechanical

Engineering are thanked fdr help in the development of the apparatus and for their care in the manufacture of the specimens, as well as for their co-operation in the day to day conduct of the testa.

The staff of Instron_Ltd. are thanked for their advice and assistance during the development of the test equipment.

The work was finances ,;by.. -thiy:.,(then)...,Department of Scientifib; and

Industrial Research. The copper was processed by Imperial Metal Industr- ies (Kynoch):.Ltd.!..and Ahe:711.11Mitium donated and processed

Industries Ltd.. The two firms are also thanked for their advice on the, problems met in the cold-working of the materials. LIST OF CONTENTS

Page

INTRODUCTION 8

2 LITERATURE REVIEW UP TO 1961 20_

2.1 Extent of Review. 20

2.2 Early development of interest. 20

2.3 Renewal of Interest of Recent Years. 25

2.5.1 Effects of cycling on the stress-strain curve to fracture. 25

2.3.2 The effect of cycling on the indentation hardness. 27

2.3.3 The stress-strain curve during strain cycling. 28

2.3.4 The stress-strain curve during stress cycling. 33

2.!3.5 Energy changes during cyclic plastic deformation. 38 ,..6 The Bauschinger effect. 40

3 DESCRIPTION OF THE APPARATUS AND SPECIMENS 45 3.1 ;general. 4.5

The Testing Machine. 45 Drive system. 45

r.2 Load weighing system 45 .:2.3 Strain measuring system. 46

..5 The Specimen and End-Fittings. 48

.3.'5.1 Choice of shape. 48

.!,.2 The. end-fitting. 50.

3.3.3 Variation of stress across the test section. 51

The Diametral Extensometer and Modified Chart Drive $y stem. 54. Page

3.4.1 General. 54

3.4.2 The extensometer. 55

3.4.3 The modified circuit. • 57

3.5- Calibration of the Scales. 58 3.5.1 Calibration of the stress scale. 58

3.5.2 Calibration of the strain scale. 60

3.6 The Strain Cycling Controls. 63

3.6.1 General. 63

3.6.2 Modifications to the crosshead cycling control system. 64 3.6.3 Plastic strain limit control. 65 4 MATERIALS AND RESULTS 67 4.1 Copper. 67

4.1.1 Condition as Supplied. 67 4.1.2 Annealed specimens.

4.1.3 Stretched specimens. 67

.4.1.4 Swaged specimens. 68

4.1.5 Cold drawn specimens. 69

4.1.6 Roller swaged specimens. 69 4.2 Aluminium. 70

4.2.1 Condition as supplied. 70

4.2.2 Annealed specimens. 71

4.2.3 Cold drawn specimens. 71

4.3 En25 Steel. 72

4.3.1 Condition as supplied. 72 -6- Page

4.3.2 Annealed specimens. 72 4.3.3 Hardened and tempered specimens. 72 4.4 Stainless Steel. 72 4.4.1 Condition as supplied 72 4.4.2 Specimens. 73 4.5 Results for Tests on Copper. 73 4.5.1 Cyclic tests between fixed limits of plastic strain. 73 4.5.2 Cyclic tests between fixed limits of nominal stress. 77 4.5.3 Other tests. 88 4.6 Results of Tests on Aluminium. 94 4.6.1 Cyclic tests between fixed limits of plastic strain. 94- 4.6.2 Cyclic tests between fixed limits of nominal stress. 94 4.6.3 Other tests. 96 4.7 Results for Tests on En25 Steel. 97 4.7.1 Cyclic tests between fixed limits of plastic strain. 97 4.7.2 Cyclic tests between fixed limits of nominal stress. 100 4.7.3 Other tests. 100 4.8 Results of Tests on Stainless Steel. .104

4.8.1 Cyclic tests between fixed limits of plastic strain. 104 4.8.2 Cyclic tests between fixed limits of nominal stress. 106

4.8.3 Other tests. 107 5 DISCUSSION 108 5.1 Survey of Cyclic Tests Between Fixed Limits of Plastic Strain. 108

5.1.1 Effect of various initial states on the cyclic behavioUr. 108 -7-

5.1.2 Effect of step changes on cyclic behaviour. 116 5.1.3 Cyclic behaviour of metallurgically unstable metal. 117

5.2 Survey of Cyclic Tests Between Fixed Limits of Stress. 119

5.2.1 Effect of equal stress limits in tension and compression. 119

5.2.2 Cycling between stress limits within the elastic range. 123

5.2.3 Effect of large values of mean stress. 127

5.3 The Bauschinger Effect and Cyclic Behaviour. 130

5.3.1 The presence of the Bauschinger effect in a symmetrical cycle.130

5.3.2 The removal of the Bauschinger effect by decreasing cycles. 134

5.3.3 The prediction of cyclic behaviour from Bauschinger tests. 138 '

5.4. Analogue Models of Cyclic Behaviour. 143

5.4.1 General. L 143

5.4.2 Models consisting of pin-jointed bars. 143 5.4.3 Models consisting of springs and sliding blocks. 145

5.4.4 Spring-block models with changing chgracteristics. 148

.5.5 Cyclic Behaviour in the Light of Dislocation Theory. 14-9

6 CONCLUSIONS 151

Table 1. AnalyAis of. Aluminium. 156

Table 2 Analysis of Steels. 156

References. 158

Volume 2 contains the Figures.

List of Figures. 163 1 INTRODUCTION . When a metal is subjected to cycles of plastic deformation between strain limits the stress range may change significantly during cycling, particularly in the first few cycles (1 ). The terms strain hardening and softening are used to describe such increases and decreases in stress range, since they are accompanied by corresponding changes in indentation hardness (2). A related effect is the progressive deformation (cyclic creep) which occurs: when ductile metalaare sUbjected to cycling between stress limits sufficient to cause plastic deformation (5). The behaviour of metals under cycles of plastic deformation is of interest to the engineer for two reasons:. Firstly, 'that he may understand the behaviour of structures in which some parts undergo cycles of plastic deformation; and secondly, that he may be able to use cyclic plastic deformation advantageously in the cold-working of metals. An understanding of the above phenomena is essential for the / development of reliable design procedure for the low endurance t<10 cycles) fatigue region. Structures which will undergo only a3 few cycles of deformation during their working life can be designed to operate with very high working stresses; which may sometimes exceed the elastic limit of the material. Such design methdds permit a great economy in weight and are of particular importance when weight is a. vital factor and design life is short (AO (3):. -9-

Not only is a knowledge of the low endurance fatigue life of a

material needed but information about the changes occuring in its

mechanical properties during cyclic deformation is also required.

Plastic deformation willocsur only in certain localities in a cyclically

deformed structure and, as many engineering components are statically indeterminate, changes in mechanical properties caused by local cyclic

plastic deformation will result in a re-distribution of the load through-

out the component. As a result of this changes will occur in the bound-

aries of the zones of plastic deformation. The design of components

to operate under such conditions preients formidable problems. Even

a, simple structure, such as a beam, when subjected to cyc]4ic bending

into the plastic range needs a sequence of calculations into which

must be fed information about the changes of mechanical properties

that occur because of cyclic plastic deformation. Even if such data

are used in idealised form the information must still be based on the

results of tests.

Polakowski in 1952 drew attention to the fact that in cold rolling

a reversal of the direction of rolling at each pass can - by imposing

a cyclic shear deformation - cause the surface of the material to be

ultimately less hard than would be the case if the passes were made in

one direction only (6). He pointed out that alternating flexure was a suitable method for producing a partial softening of materials that

had been work-hardened by uni-directional deformation during the ordinary processes of metal working. -10-

More recent tests on specimens subjected to cycles of plastic deformation and, at the same time, a small steady superimposed stress have shown that the latter could produce large deformations (7). This has led to the development by Coffin of the contact-bend-stretch rolling process ( 8). In this a four high rolling mill, with the addition of a small diameter floating roll is used. This is shown in diagramatic form in Fig. 1. The floating roll, which recieves support from the two large contact rolls, can be of very small diameter and thus impose a large bending stress. The strip is bent four times as it passes through the mill; the greatest bending stresses being imposed on the second and third occasions. Each bending coincides with the passage of the strip between a pair of rolls and throughout the process tension is applied. Accurate rolling to thin gauges with large reductions per pass can be achieved with this mill. The process,can reduce the hardness of a material that has been initially severely work-hardened and a reduction in thickness of as much as 98% has been achieved without annealing. The effectiveness of the process clearly depends in part on the cyclic plastic deformation imposed by the bending. So many variables are present that if these are to be optimized for each task then a theoretical approach based on material behaviour is desirable.

Material that undergoes cyclic plastic deformation during service is usually subjected to a complex system of stress; and in any cold- working process in which cyclic plastic deformation is used the stress conditions will not be simple. It is clear that any fundamental -1i-

investigation into the behaviour of metals under cycles of plastic

deformation must be based on a study of behaviour under simple conditions of stress. Only when behaviour under these conditions is understood can an attempt be made to use the experimental obser-

vations as a basis for the interpretation of behaviour under condit-

ions of complex stress. With cyclic deformation a wide range of possible stress systems

exists. Cyclic stress systems may be classified as follows:-

(1) Uni-axial tension and/or compression along a fixed axis.

(2) Bi- or tri-axial tension and/Or compression on fixed principal axes.

(a)With proportional variation of the stresses.

(b)With non-proportional variation of the stresses.

(c)With one or two of the stresses•held constant while the others

vary.

(3) Any systems similar to the above but with principal axes which do

not remain fixed in direction during cycling.

Some of the simpler forms of cyclic loading test produce some

of the more complex cyclic stress systems. For example; a thin cylinder

acted on by a constant axial force together with a cyclic torque gives

one of the more complex cyclic stress systems (classified under (3) above)

because the principal axes of stress do not remain fixed in direction

during cycling. It so happens that some of the more complex stress

systems are easily produced in the laboratory while the simple stress

systems soMetimes present great difficulties to the experimenter. As• -12-

a result of this a great deal of the earlier research work in this field was done using complex systems of stress.

If an investigation is to be restricted to one of the simple

stress systems then the choice must be made between uni-axial tension

and/or compression tests or pure shear tests. (The former is classed

under (1),above and the latter is a special case of class (2a) above.)

It is difficult to produce stresses which approximate tc pure shear • under conditions of large deformation - a thin walled cylinder will

develop local buckling at quite small plastic strains. Thus - in

spite of the practical difficulties involved - a tension-compression

test is the most suitable for use in an investigation into the behav-

iour of metals under cycles of plastic deformation.

Until the last decade little research had been done on the behav-

iour of metals when subjected to tension-compression cycles of plastic

deformation. Undoubtedly the main reason for this is that the necessary

apparatus, in particular extensometers of high magnifying power, has

not been available. The design of tension-compression machines does

not present a very difficult problem and in the early years of the

century Dalby converted a hydraulic tension testing machine for use

as a tension-compression testing machine. In tension-compression

testing it is advisable to use a waisted specimen because a long cylindrical specimen will buckle under compressive load. The problem of

the design of an extensometer for use on a specimen having a very

short test length and suitable for use with an autographic recorder -13-

was, until the advent of electronic amplification, almost insurmountable.

Dalby developed his extensometer and photographic recorder for use on

a specimen having a short parallel gauge length. The results he obtained

( 9), though of great historioal.interest, were of very poor quality.

As the apparatus he used was fat in'advdnce of contemporary autographic

recording equipment it is not surprising that a long time elapsed before a further attack was made on the problem.

Of recent years tension-compression testing machines designed for

low endurance fatigue testing have becoke available. A machine designed

to operate in both tension and compression and provided with automatic cycling controls operated by the load measuring system allows tests to

be made between fixed limits of nominal stress. Various workers in the field of low endurance fatigue (V) (11) have designed strain

measuring and recording equipment for use on their specimens and have thus been able to investigate the changes that occur in the shape of the stress-strain loop during cyclic plastic deformation between fixed limits of nominal stress.

In practice the case in which cycling occurs between fixed limits of total strain is of greater interest. This is because the most highly stressed parts of a structure are often small regions in which the deformation is controlled by the elastic deformation of the main mass of the structure. If cycles are. to be made between fixed limits of strain it is necessary that the extensometer signals shall control the reversal of the machine. Ideally the rate of application of stress or strain as well as

the limits of stress or strain should be oontrolled. For example, in

a structure in which the load is increased at a constant rate the elastic

behaviour of the main mass imposes a constant total strain rate of deformation on the most highly stressed parts. If a'material is to be tested

under such Conditions the extensometer:must control the:machiri.e through-

out the cycle and not simply.effect the reversals.when the required. limits

of. strain are reached. To do this makes even greater demands on the

quality of the extensometer'and calls for an elaborateautomatic control

system in the machine. Equipment of this kind is being developed but the

present practice is to accept the strain rate variations imposed by the characteristics of the testing machine. In the case of a machine with constant cross-head velocity the conditions are approximately those of constant stress rate. This was the case. in the present work.

The present study of the behaviour of metals when subjected to tension- compression cycles of plastic deformation was undertaken after a review of literature - see Chapter 2 - had shown that there was little information on the effect of strain range, strain history or previous heat treatment or, the cyclic strain hardening and softening. Such information as was available had often been obtained from stress-strain observations made during low endurance fatigue tests. Such tests are conducted at a single amplitude of total strain and - because of the very short life with large plastic deformation, the information that exists,is:'mainly.for the region in whicir.the'.elastiC strain forms an appreciable part of the total strain. -15—

It was decided to confine the present study to room temperature conditions. Thus permitting more tests to be Made than would be possible if other temperatures were used; and hence making it possible to investigate some of the materials for a wide range of initial states.

Oxygen free high conductivity (OFHC) copper was selected as the material to be used for the majority of the tests. It was chosen because it is commercially available in a fairly pure form, has a large capacity for work hardening, and is of practical interest in that it is used in the cold worked state in the electrical industry. A study of its cyclic properties had been made by Benham in the course of an investigation into its fatigue and fracture behaviour (514.) and, under his guidance, prior to the inception of the present investigation, the author had made a further study of the effects.of previous.strain history on cyclic deforMation by subjecting copper cubes to cyclic compression (50).

Commercially pure aluminium - which, like copper, has a face centred cubic structure - was also selected for testing.

It was also decided to test a nickel-chrome-molybdenum steel (En25).

This was chosen because.it could be given a wide range of initial hardness values by heat treatment and was known to be metallurgically stable.

Fully soft 18-8 Ti austenitic stainless steel - known to be metallurgically unstable - was selected for comparison with the En25 steel.

It was decided that the strain ranges should extend from about

0.170 to 5.0,4.. This covers the field that is of interest both to those designing parts to have a life of less than 1000 cycles and -16-

to those concerned with the use of cyclic deformation in metal working processes. It was not intended to continue tests beyond 1000 cycles.

Plastic strain limits instead of total strain limits were chosen for use in the strain cycling tests. This was done because the changes in mechanical properties depend on plastic deformation and not on elastic deformation.

The test programme for copper was as follows:-

Annealed copper specimens were to be tested at various amplitudes of plastic strain. A similar series of tests were to be made on copper which had been initially work hardened to various degree a.

It would have been preferable to use simple tension as the sole method of cold working the copper but this would have limited the hardness so produced to a low value. Therefore it was decided to use a variety of methods of cold working thus enabling higher degrees of hardness to be produced. Watts and Ford (t3) have'shown that tests' on cold rolled copper conform to the basic stress-strain curve for the material. Thus it is reasonable to expect that comparative cyclic tests made on copper that has recieved the same deformation by various methods of cold working will also show agreement. If this is so the investigation can cover the study of the cyclic hardening and softening of copper having a range of initial hardness from the fully annealed state to almost the maximum hardness obtainable.

If - as has been suggested by some investigators - copper tends to the same cyclic state when cycled at & fixed strain amplitude -17-

irrespective of the degree of initial cold work then it may be supposed the stable cyclic state is independent of the previous strain history of the metal. To further investigate this it was proposed to teat each of a set of annealed copper specimens at a series of strain amplitudes; the cycling at each amplitude being continued until the stress amplitude tended to a limiting value. The results of these step-tests could then be compared with those for a. single amplitude of cyclic strain.

If the stable cyclic state is independent of the previous strain history then it follows that cycling between stress limits should give the same stable condition that is produced by cycling between the corresponding strain limits. To investigate this it was proposed to carry out a number of cyclic tests between fixed stress limits.

When cycling between fixed limits of stress progressiver. ftUd!or.._ mation (cyclic creep) is unavoidable. If this is large the change of shape of the specimen results in experimental difficulties of a kind not encountered when cycling between fixed limits of.plastic strain. Because of this it was decided to condUct,041Y.A investigation into the behaviour of copper when.eyeleYbetfeen fixed limits of stress.

.The key to the prediction of the behaviour of metals under cycles of plastic deformation may lie in a better understanding of the form of the stress-strain curve for reversals of stress. It was therefore decided to carry out reversed loading tests on annealed copper.and on .48-

copper that possessed.,various histories. of paild-yprke including some forms of cyclic deformation. The test programme for aluminium was as follows:- Annealed aluminium specimens were to be tested at various amplitudes of plastic strain. Similar tests were to be made on aluminium" in the cold drawn state. The test programme for Eri25 steel was as follows:- Annealed steel specimens were to be tested at various strain amplitudes., Similar tests were to be made on steel which had been initially heat treated to give various degrees of hardness. As this steel has only a slightcapicity for work hardening in tension it . was not proposed to carry out extensive tests on the material in the cold worked state. .1t,wasAlecided_toaakesome reversed loading tests on both the annealed and heat treated steel. The test programme for Stainless steel was as follows:- Specimens of steel in the fully softened condition were to be tested at various strain amplitudes. Further test programmes:- If a cyclic test is interrupted at any point in the cycle of deformation and s• tension or compression test then performed it is not to be expected that the tension and compression tests will have stress-strain curves. of identical shape. It is interesting to know if this asymmetry can be removed by a suitable programme of cyclic deformation. It was proposed to investigate the possibility of -19-

using a, sequence of cycles of gradually decreasing strain amplitude to restore symmetry to the tension and compression characteristics of specimens that have undergone cyclic deformation. -20-

2 LITERATURE REVIEW UP TO 1961

2.1 Extent of Review

The review extends to the end of 1961; at which time the present research

was commenced. Papers of later date are dealt with in Chapter 5 where

they are discussed in relation to the present work.

2.2 Early Development of Interest

An interest in the cyclic plastic deformation of metals has existed for

more than a century. It appears to have arisen as a result of inves-

tigations into the effect of the cold working of metals on their magnetic

properties. The existence of & hysteresis loop in the load-deflection

diagram for cycles of plastic deformation led the workers concerned to

compare these loops with those observed during cycles of magnetization.

Todhunter and Pearson refer in their History 02) to an investigation by

Wiedemann in 1860 into the behaviour of annealed mild steel and brass

bars subjected to cycles of plastic deformation in torsion and a similar

investigation using cycles of bending. As the bars were solid the res-

ults were only of qualitative interest because the stress distribution

throughout the material was not known. They do, however, show the exis-

tence of cyclic strain hardening phenomena as revealed by the changes

in form of the hysteresis loops:. The word 'hysteresis' was not then

use& - it was coined by Ewing, from the Greek 'coming after', and first appeared in print in 1881 (10. Reversed torsion tests on solid or thick-walled specimens are per-

haps the easiest kind of cyclic test to perform, for the specimens used -21-

can be of reasonable length without there being a risk of them buckling.

Unfortunately the reduction of the wall thickness to give a close approx- immtion to a uniform, and therefore a known, state of stress re-introduces the problem of instability, for such specimens are liable to suffer local

buckling. ,However, 'though the results of tests made on solid specimens

may (in the present state of knowledge) have no quantitative value they can be a, source of much qualitative interest. They can, for instance,

be used to show the existence of a, l.imiti.ng value of hardness for cycles

of plastic deformation (15).

It was the problem of designing parts to withstand fatigue - that

is ordinary, long life fatigue - that gave the impetus to the first

quantitative studies of cyclic plastic deformation. The driving force. • was the desire to find some form of short test that could be used to

predict the endurance limit of a material. The problem of fatigue was

first discussed by Poncelet; probably as early as 1839. Bauschinger,

in 1886, described a series of tests he had made in which specimens were

deformed alternately in tension and compression 16). Each plastic def-

ormvziozi wrIs very small and it waa roonihla rarry out the tests on

pecAmens having a parallel portion to which a very sensitive mirror*

eytensometer could be attached. The specimens were subjected to several

reversals of stress. The accuracy of the observatior was such that

Bauschinser was able to observe the changes that occl,red in the limilAs

of proportionality and to investigate the effects of rest periods on

these. He established that there is a 'natural limit of elasticity' -22-

to which the material tends after a number of cycles. He sugmested

that this value decided the endurance limit of the material in fatigue.

Bairstow (17) extended the range of Bausohinger's investigations

using similar methods of testing. He discovered that a 0.34c/0 C steal

(having a yield stress of 24.9 tons/ins) which appeared. at first, per- + fectly elastic when stress cycled at 14.1 tons/In2. developed a small

amount of cyclic plastic deformation after 18 750 cycles. Nis tests

showed that this cyclic softening of the material occured when the ampl-

itude of the cyclic stress did not exceed the stress at which the yield

step occured in the original material. He observed no change in the

mean length of the specimen when using equal values of tensile and com-

pressive stress. However, his plastic strains were very small and,pres-

umably his stress settings very precise. He also carried out stress

cycling between limits that were both tensile and showed that so long

as the stress-strain diagram gave a hysteresis loop a progressive inc-

rease in the mean length of the specimen occured.

The plastic strain amplitudes. measured by both Bauschinger and

Bairstow were very small compared to the total strain amplitudes. Early

investigations in which specimens were subjected to larger plastic def-

ormations were limited to only a few reversals of stress because the

parallel portions of the specimens used tended to fail by buckling

under comnressive stress.

Grmy, in 1897, fitted an extensometer to a mild steel specimen which

had a. gauge length of 3 in. and au diameter of 1 1/8 in.. The extensometer -23-

was kept on the specimen and the latter remounted in the testing machine each time it was necessary to change the direction of loading. A stress-

strain curve for three reversals of stress was recorded autographically M.

The greatest strain applied was about 1%. His tests suggested that,a

single reversal of strain within the yield step region was not sufficient to remove the yield step but that a reversal after the material had been deformed beyond the yield step resulted in the disappearance of the sudden yield phenomenon.

About 1910 Dalby (9) converted a machine for tension-compression testing; thus avoiding the necessity of having to remount the specimen in the machine each time the direction of loading was reversed. He also

used what was in effect, a load cell for measuring the load instead of

the usual lever system. The values of both-load and strain were trans-

mitted to a. mirror system in an autographic recorder and the stress-

strain curves recorded on a photographic plate. Compared to the apparatus used by other investigators at this timeDalby.'s apparatus must

have been capable of a fast rate of straining. The use of an axial

extensometer necessitated the provision of a parallel portion in each

specimen. Only a small number of cycles could be made and there vas

no provision for automatic reversal of the machine during testing. His

tests - which were an extension of his work on repeated plastic defor-

mation in tension - like Gray's, showed that a plastic deformation bey-

ond the yield step resulted in the absence of a. yield step during immedia-

tely following deformations. Ludwik and Seheu in 1923 (19) in load cycling tests on torsion specimens of copper, aluminium and steel showed the tendency of annealed metals to harden towards a. limiting value of hardness under cyclic deformation and work hardened specimens to soften towards a similar value and that the approach to the limiting value was rapid when the cycles of plastic deformation w*re large. During the final cycling preceding fatigue failure they observed an apparent softening to take place: and they also observed the reappearance of a yield step (which previously appeared only in the first quarter cycle). It is of interest to note that Burns in 1966 (20) reports that annealed EN25 steel which had softened towards an apparently settled cyclic state showed a final hardening towards the end of its fatigue life (an effect which was not caused by profile changes). The development of cracks might explain an apparent softening but the apparent hardening would be reduced by by the propogation of cracks.

Bauschinger's research had shown that hardening produced by uni- directional deformation could be reduced by a reversal of the deformation. Armour in 1930 heat treated 0.35% C steel to various degrees of hardness and showed that a t5% reduction of area by drawing caused an appreciable reduction in the hardness of the steel (21). Mc Adam, Geil and Jenkins-in4947. (44)' extended Bauschinger's work by testing specimens in tension or compression in a direction at right angles to that of the original deformation. All these tests, although confined to less than one oycle,showed that plastic deformation can cause softening. -25-

2.3 Renewal of Interest of Recent Years The revival of interest in the subject of cyclic plastic deformation of recent years has been largely due to its relevance to the problems of low endurance fatigue (4) and metal working (8). Changes in the mechanical properties that occur during cycling can be observed in

the following ways:- 2.3.1 Effects of cycling on the stress-strain curve to fracture. Polakowski in 1952 (6) subjected specimens to a few cycles of plastic deformation

by bending them round a mandrel. The metals tested: aluminium and copper of commercial purity, HC copper, aluminium-1.2% magnesium alloy and stabilised low carbon steel; were selected because of the absence of precipitation hardening effects. Cold work promotes ageing effects and the behaviour of metals under cycles of plastic deformation would be complicated if ageing was induced in the material. Materials were .tested in the annealed state and also after cold drawing or rolling by various amountsl(notexceeding 60% reduction of area). Tensile tests to destruction were made after the completion of a series of cycles • of plastic deformation and the stress-strain curves recorded. He reports that in the case of cold worked materials afew (less than 30) cycles could so soften the metal amto make its peroentage elongation at fracture twice its initial value. Polakowski% results are only of quantitative value because the whole of the material was not subjected to uniform cyclic deformation. Although it was not practicable to

carry out a large number of cycles of plastic deformation the tests -26- suggested the existence of a settled cyclic state for each radius of bending. He points out that this behaviour may explain the fact that there is fittle difference in the fatigue life of a metal when tested in the annealed and in the cold worked conditions. Polakowski and Palchouduri made a further investigation into the softening of cold worked metals in 1954 ( 2). The metals tested were: copper, nickel, aluminium, three alloys of these metals and a non- ageing titanium killed steel. The specimens were subjected to uniform cyclic deformation in an Ameler Vibrophore tension-compression fatigue machine. The use of this machine made it possible to apply a large number of cycles but made it necessary to restrict the amplitudes of the deformations to small values. After cycling test pieces were cut from the material for use in compression tests. The results of these tests gave a measure of the hardness of the material. Compression tests•. were used in preference to tensile tests because the presence of any cracks which might develop would influence the results of tensile tests. more than those of compression tests. Despite the large number of cycles used it was not possible to come very near to the settled cyclic state; presumably because of the man strain amplitudes used. Never- theless, these tests:gave furtherevidenoe.thata,.settled cyclic state probably existed. 6 Broom and Ham in 1957 (22) also made cyclic tests (up to 10 cycles) on copper at room temperature and at a low temperature and then tested the specimens in tension. They state• that the form of the stress-strain -27-

curve after cyclic deformation is dependent on the the temperature

at which the cycling and testing were carried out,, They suggest that

point,defects are responsible for hardening in cyclic deformation

because these are known to suffer a decrease of mobility when the

temperature of the metal is. lowered. The Amsler Vibrophore machine

was used for these tests and they point out that the design of the

machine does not permit the, desired stress limits to be achieved

immediately at the start of a test.

2.3.2 The effect of cycling on the indentation hardness. Polakowski ( 6)

and Polakowski and Palchouduri (2 )in the work described above made

indentation hardness tests in addition to the tension or compression

tests after cycling. The changes in indentation hardness agreed with

the changes as indicated by the other tests, as would be expected.

They did not, however, take advantage of the fact that indentation

hardness tests can be made at intervals to measure the progressive

change of hardness occuring during cyclic deformation. Polakowski

in 1951 (23) did make a series of indentation tests on a specimen

which was subjected to a single reversal of plastic deformation but

as these investigations were not concerned with multi-cycle deformation

they are dealt with later.

Indentation hardness'tests made on the surface of a solid specimen

during torsion or bending can be used to measure changes of hardness

there during cyclic testing. Thus they can give quantitative results

for tests made on solid specimens because the surface strain (though -28-

not the stress) can be calculated. Despite the objections to the use

of an indentation test to measure the plastic properties of a material

the simplicity of the cyclic torsion test when applied to solid specim-

ens makes this form of test very attractive. It is remarkable that

tests of this kind have not proved popular in the earlier work on

cyclic hardening and softening.

2.3.3 The stress-strain curve during strain cycling. The stress-strain

curve traced during a cycle provides much of the information given by

a test to destruction but without the necessity of terminating or

even interrupting the cyclic deformation. The study of it is, therefore

of great value to workers in this field. From it the changes in the

limit of proportionality can be observed - though this quantity is

so ill defined as to be of little value. Various: proof stresses can

likewise be read from the curve. The larger the specified strain used

for their determination the more accurate the observation.

It is convenient when cycling between fixed limits of strain to

use the value of the .strain amplitudeto define the proof stress. It

is then only necessary to observe the peak values of the stresses that

occur in each cycle. The magnitudes of these are then used as a measure air of the hardness. This has been done by many investigators and the

method has the added advantage that the observed quantities are of

practical value to designers.

Wood and Davies in 1953 (21f) described the effect of a few tension-

compression cycles of plastic deformation between fixed limits of strain -.29-

on electrolytic copper. Their main interest lay in observing the

changes of structure caused by cycling but they did record some stress-

strain loops and they plotted the values of the peak stresses on a

basis of cumulative strain (summed without regard to sign). This

method of presentation showed that the hardness produced by cyclic

deformation is less than that produced by the equivalent uni-directional

deformation. It also showed that the smaller the strain amplitude

the less the hardening. Their results do not indicate a limiting

value of hardness for each strain amplitude, but their tests did not consist of many cycles. They also did a test in which the cyclic

strain amplitude was increased in steps, several cycles being carried

out at each amplitude. The hardness that was observed to develop at each strain amplitude during these progressive tests was less than that observed when the material was cycled at a constant strain amplitude. The comparison between step and constantramplitudertesta • provides little useful information because of the small number of4.

cycles used.

Coffin in 1954 studied cycling under thermal stresses using an apparatus (25) in which the ends of the specimen were held by a rigid frame while the specimen temperature was cycled. Thus the specimen underwent simultaneously cycles of plastic deformation and cycles of temperature change. The study of such thermal stress cycling is of . great practical interest but the test conditions are so complex that the results can give little help in the understanding of plastic -30- deformation in general. However, Coffin extended his investigation in various ways, on of which was to impose plastic strain on the specimen by external means. This was done by cycling the temperature of the columns which formed the sides of the frame which held the specimen.

Some of these tests, made om a specimen held at constant temperature, were reported by Coffin and Read in 1956 (26). The specimens tested were of Type 347 stainless steel. The tests were made at both room and elevated temperature at a strain amplitude + of 0.5%. Hollow cylindrical specimens were used because this form gives a good resistance to buckling. A gauge length was marked on the parallel portion and a microscope was used to measure the changes that occured in this. Teats were made on fully annealed specimens and some were made on specimens that had been hardened by cold-working.

The cold-working was done by twisting the specimens. Such a method of cold working has the adventage that it does not introduce asymmetry between the tensile and compressive properties as observed in the cyclic tests. This simplifies the investigation in that it separates the* effect of hardening or softening from the effect of the elimination of asymmetry of properties; both of which arise from the application of cycles of plastic deformation. However, the use of one kind of uni-directional deformation followed by some other kind of cyclic deformation can hardly be looked on as a simplification of the basic problem.

Wood and.Segall in 1957 (27) tested copper, nickel, aluminium and a brass in the annealed state in cyclic torsion using various strain amplitudes. They used solid specimens with their attendant disadvantages. They chose this form of test in order to be able to cover a wide field of investigation in a short time. Their work was concerned with the structural changes occuring in the material, but they recorded the torque amplitude when cycling between fixed limits of angular twist and plotted the results on a basis of cumulative twist.

They continued the tests to total deformations of as much as 3 500% in the case of small amplitudes. Although, because they used solid specimens, the results are qualitative, they showed the existence of definite limits of hardness for various amplitudes of cyclic deformation.

In 1958 (15) they reported a similar series of tests on several of these metals in various states of cold work. An extensive investigation was made in the case of OFHC copper. A wide range of strain amplitudes were used. The cold work was done by stretching, drawing or swaging but the cyclic tests were made in torsion. Thus the comments made above on Coffin's method of cold working his specimens apply also to the work of Wood and Segall. Copper work hardened by a 7O reduction of area and cycled at various strain amplitudes showed a small initial increase in hardness and then softened towards a limiting value of hardness appropriate to the amplitude of deformation used.

Copper reduced by 10, 20 and 30% reduction of area by stretching fol7- owed by cyclic tests at a particular (small) amplitude of deformation -32- also showed a small initial hardening followed ty softening; apparently towards a limiting value. Coffin and Tavernelli in 1959 (28) reported an extensive investigation'into the behaviour of several metals subjected to tension- compression cycles of strain. They used a waisted specimen with a radius profile at the neck. The diametral strains were observed by means of a dial gauge type extensometer which calipered the neck.

The cyclic testing was done in a tension-compression type of testing machine using manual control for the reversals. During the cycles in which stress-strain curves, were plotted the speed of the machine was reduced to allow a number of observations to be made.

The metals tested were: 2S aluminium, OFHC copper. SAE 1018 C steel, AISI type 347 stainless steel, nickel, titanium and two alloys of aluminium. All the testa were at room temperature and the strain + amplitudes used ranged from 0.2% up to 50% axial strain. In some cases the tests continued to several thousand cycles but in the case of aluminium geometrical instability limited the number of cycles to about 200. The occurence of geometrical distortion was: later investigated by,Coffin (7) (29) who showed that it accounted for an apparent increase of hardness during the final cycles on some of the specimens.

The hardening and softening observed was, in the case of most of the metals tested, similar to that observed by Wood et al. (15) for cyclic torsion. There,is, however, some doubt as to whether a small initial hardening occurs prior to the softening of cold worked metals. In -33-

testa where the pre-straining is done in tension and the cyclic strain amplitude is small it is not to be expected that tension-compression symmetry will be produced by the first few cycles; thus, if one curve is drawn through the points on the graph of stress amplitude against the number of cycles the points will lie alternately high and low. Further, if, following tensile pre-strain, the cyclic testing commec- ces with a compression the Bauschinger effect will cause the first point to be lower than the second and this will suggest that there

is an initial strain hardening. The results for metals cold worked to different degrees and then cycled at the same strain amplitude showed that they tended towards the same degree of hardness, as had been suggested in Wood's torsion tests. The tests on stainless steel showed that after a certain number of cycles a remarkable .increase in hardness oceuredl"bhia being obser- • ved to occur earlier when the amplitude of the cycles was increased. This it attributed to a_ metallurgical change occuring in the steel when subjected to cycling. It is interesting to notErthat Franz in 1954 (30) showed by repeated tensile straining of:a, preoipitation_ • hardening metal (75S-T6 aluminium'alloy) that:repeated:deformation can cause a marked decrease of ductility; an effect not:obseried in a non-precipitation hardening metal (copper). 2.3.4 The stress-strain curve during stress cycling. no boirbietialirriat metals whet cycled between fixed limits Of..strailt:ham:beet studied -34--

both for the case in which no reversal of loading occurs and the ease

in which a change in sign•of the load occurs:during each cycle. Tests

in which complete or partial removal of the load takes place form a

, natural extension to the study of the creep behaviour of metals under

constant loading. Tests in which reversal of the load occurs can involve a reversal of plastic strain as well as a progressive' change in the

mean strain during each cycle. Provided the load limits are both tensile it is possible to carry

out investigations using an ordinary tensile testpiece and a tensile

testing machine with autographic recording equipment. Of recent years

machines have become available in which automatic reversals can be

made to occur at preset values of the load, thus permitting automatic

cycling between fixed limits of nominal stress. The extension of

investigations into the wider field in which-stress reversal may occur

has had to await the development of the kind of test equipment used

in the study of strain cycling. • a

When ductile metals are subjected to cyclic deformation in the

presence of a steady stress a progressive deformation (cyclic creep) may occur with each cycle. This effect has been observed to occur in

tests made under conditions of both uni-axial and of coioplex stress Its

occurence under conditions where the steady stresals very small have been noted by Coffin, who pointed out that the ball contact of a

dial gauge can indent into the surface of a specimen that is under-

going cycles of plastic deformation (7.). In tthis:-instance a. state'. of -35- complex stress is present; the steady stress is of one kind and that which causes the cyclic plastic deformation is of another kind. Coffin investigated this phenomenon in an experiment in which tension-comprosaion cycles of strain were applied to a: specimen which was, at the seam time, subjected to a constant shear stress produced by torsion (7). fie showed that large deformations can be produced by small stresses when the material undergoes cyclic plastic deformation. It is not possible to determine from his results whether the cyclic creep tends towards a steady value because the tests were made on solid specimens.

A special case of cyclic creep can be observed when uni-axial tension-compression tests are made between fixed limits of nominal stress. In this case the value of the mean stress during cycling may be lookad upon as the 'steady stress' while the variation of the applied stress from the mean may be treated as the fluctuating stress that causes the plastic deformation.

The progressive change in dimensions of specimens during low endurance fatigue tests on copper cycled between fixed limits of nominal stress was observed by Benham (31)(10). He*showed that, under certain conditions, a creep type of failure could occur instead of the fatigue type of failure characteristic of long life tests. Many of the tests were made at zero mean nominal stress and under these conditions a progressive elongation of the specimens was observed.

It appeared that this might be explained by the fact that when the mean nominal stress is zero the mean true stress must have a small -36-

tensile value - the result of variation of specimen cross-section during each cycle. To study this he made small alterations in the value of the mean stress by changing the value of the tensile stress limit during a test. It was found that an alteration of as little as

30 lb (on a range of 6 tons) could reverse the direction of the cyclic creep. So sensitive was the direction of cyclic creep to the value of the mean stress that only by constant adjustment of the load limits could cyclic creep be prevented - such a procedure being, in effect, a reversion to strain cycling. He attributed cyclic creep to instability of the material but stated that, for given stress limits, the material tended towards a steady state of cyclic creep. Stress cycling tests present certain practical difficulties not encountered in strain cycling tests. The changes that occur in the cross-section of the testpiece are the major problem. With a ductile metal like annealed copper quite a small stress amplitude will cause a large change in dimensions on the first application of load; though after many cycles the plastic deformation may become quite small. It is therefore preferable to carry out tests using limits of true stress rather than limits of nominal stress. If limits of true stress are to be used the load at which reversal is to occur must be calculated afresh from the current area of cross-section each time it is required.

Thus cycling between limits of true stress: is, at present, only practicable if the machine is operated at a very slow speed and controlled manually. Further progress in this field awaits the development of -37-

electronic equipment capable of computing and recording the values of true stress.

Coffin showed in tests on waisted specimens cycled between fixed limits of strain that a progressive change of profile geometry occured during cycling (28)(7). If, as in his' tests, the diameter at the waist of the specimen was controlled then a gradual increase in the diameters at all other sections occured; the larger the local di,,meter the greater was the rate of. growth. lie explained this behaviour by the fact that only at the test section was the mean true stress of the correct value; while at larger cross-sections its value was such as to cause a progr essive local shortening and hence an increase in diameter. He shoWed that the effect is most pronounced in aluminium because it has only smell capacity for cyclic strain hardening.

This geometrical instability effect is present whatever kind of cycling is applied to the specimen. In a waisted specimen in which the profile is formed by a circular arc the effect is to decrease the radius of curvature. In a cyclidrical specimen the instability shows itself in the development of waves in the surface.

Coffin showed that severe distortion of the profile in a waisted specimen - as may occur in aluminium - results in an apparent hardening of the material towards the end of its life. This ifs a spurious effect caused by a change in the Bridgman factor (32) resulting from an alteration in the profile radius.

GeOmetrical instability is a• result of cyclic creep and in his -38-

study of the former Coffin investigated the cyclic creep properties

of several materials. These included: 23 aluminium. OFHC copper,

low carbon steel, nickel, 34.7 stainless steel, all in the annealed

state; OFHC copper swaged to 33% reduction of area; and aluminium alloy

24.ST in the stress relieved condition. To avoid the difficulties encountered in the study of cyclic creep

by cycling between fixed limits of stress Coffin used a programme of

strain cycling in which the limits were altered from cycle to cycle.

The tensile strain in each cycle was made to exceed the compressive

strain by a fixed amount. Thus the specimen was subjected to cyclic

deformation while a steady rate of cyclic creep was imposed on it.

The peak tensile and compressive load values were measured in each

cycle and from these the peak values of tensile, compressive and mean

true stresses were calculated. Manual control of the testing machine

was necessary but this method avoided the calculations needed when

cycling between limits of true stress. By so doing he also avoided the

difficulties presented by the occurence of very large strains during

the first few cycles when using fixed limits of stress.

2.3.5 Energy changes during cyclic plastic deformation. The study of the

energy used during fatigue testing has always been of interest to

engineers. By means of temperature measurements made on specimens

during test comparative values for the energy consumed per cycle can

be obtained. Thus, long before the development of delicate strain

measuring and autographic recording equipment, there have existed - 9 -

sensitive means of detecting small changes in the amount of energy consumed per cycle.

Calorimetric studies made by Stromeyer in 1914 (33) and further

work by Gough in 1923 (34) made on medium carbon steel and using torsion

specimens indicated that this method provided a means of predicting the value of the limiting range of fatigue stress by testing a single specimen. They were, in effect, observing the widening of the stress-

strain loop that had been reported by Bairstow in 1910 (17).

Inglis in 1927 measured the total work to fracture for rotating

specimens tested in bending (35). He found that the total work to fracture a specimen decreased as the stress was increased.

The problem of fatigue failure by cumulative damage led to a revival of the energy theory of failure. Miner in 1945 (36) proposed a rule for use by designers based on the hypothesis that the total energy consumed by a specimen was a measure of the cumulative damage that it had recieved. By the use of this rule designers could use the results for tests conducted at constant stress amplitudes for the design of parts subjected to sequences of alternating stress at various amplitudes.

With the development of the study of low cycle fatigue it is natural that similar attempts should be made to correlate low endurance fatigue life with the total energy consumed. In this case the work done per cycle is large and calorimetric methods cannot be used. It is necessary to derive the amount of work from the areas of the -40-

stress-strain loops.

Benham (31) and Benham and Ford (10) in 1961 investigated the relat-

ionship between low endurance fatigue life and the plastic strain energy

to fracture in tension-compression tests made both between fixed limits

of nominal stress and fixed limits of strain. Their results showed

that the same relationship existed between total energy to fracture

and number of cycles to failure for both kinds of cycling.

Tests were made by Feltner (37) in 1961 on steels cycled in the

intermediate range between high and low endurance fatigue life; in which

anelastic strain energy is an appreciable part of the total strain

energy consumed per cycle. He pointed out that only the energy expended

in plastic deformation and not that concerned with anelastic behaviour

should be considered when studying the fatigue life of metals. 2.3.6 The Bauschinger effect. Bauschinger in 1887 (16) summarised the r results of his own research into the effect of plastic deformation

in tension (or compression) on steel when subsequently tested in

tension or in compretsion. - If the yield point has been exceeded in

tension then immediate reloading in tension shows that the limit of

proportionality it depressed almost to zero, but the yield point is

raised approximately to the maximum stress applied in the initial

loading. Rest or heat treatment cause the limit of proportionality

to rise to above its original value and the yield point to rise slightly

above its previous value. If the yield point has been exceeded in

tension then immediate reloading in compression shows the limit of proportionality to be very much reduced, while there is now no distinct yield point. Rest now causes onlia very gradual change in the limit - of proportionality. The behaviour of a metal that has been initially strained beyond the yield point in compression show!) the corresponding

phenomena. The markedly different behaviour of a metal when after plastic deformation in tension (or compression) it is tested in compression (or tension) has come to be known as 'the Bauchinger effect'. The expressior is given & broader meaning by Lubahn (38), who applies it to the effect of plastic deformation in tension (or compression) on the behaviour of the metal in any kind of subsequent test - including a repetition of the original loading and including creep tests. Bauschinger extended his testa to investigate the behaviour...of metals under several reversals of stress. He thawed that with a stress range sufficient to cause appreciable plastic deformation the limit of proportionality at each successive application of load imooses lower

and lower and rapidly approaches zero. He then cycled the sOerial between much lower limits of stress and showed that, with suitably chosen stress limits, the limits of proportionality, in both tension and compression, will rite together toward* thirsame maximum value. These new limits he called 'the natural elastic limits'. (Unfortunately, it appears that in Bauschinger's papers the term 'elastic limit' is treated as synonymous with 'limit of proportionality'; though it is,

clearly the latter that he was measuring.) When the material is cycled between the natural limits its deformation is entirely elastic. If

either limit is raised by overstraining then the other is lowered.

If the amount of overstrain is large it is possible for both limits

to have the same sign.

Polakowski in 1952 (39) pointed out that as a result of the Bauschinger effect it is possible to soften cold worked metals by

the application of a small amount of reversed plastic deformation

and that this effect might provide an explanation for the phenomena

of the softening of metals under cycles of plastic deformation. He

investigated the Bauschinger effect for normalized steels of 0.28 and

0.44% carbon content by making compression tests on cylindrical spec-

imens that had been cut from tensile specimen's which had recieved an

elongation of 10.57. Vickers indentation hardness tests were made on the cylinders at intervals throughout both the tension tests and

the compression tests. These show that the rise in hardnesa during

the tensile deformation is followed by an initial fall in hardness

during the compressive deformation. At about 11'7; compression the

hardness reached a minimum value and at 30% compression it had almost reached its original peak value.

Attempts. have been made to explain the Bauschinger effect on

the grounds that poly-crystalline metals are composed of grains which are anisotropic in their mechanical properties and which do not all

have the same orientation. Thus when the metal is subjected to an increasing load some of the grains will suffer plastic deformation

before others. Removal of the load will leave a system of internal

stresses within the metal. If the metal in its original condition

has the grains orientated entirely at random - and this is not always

the case in annealed metals - it will be isotropic in its mechanical

properties. After overstraining the presence of the internal stress

system will cause it to be anisotropic and it will not give the same response to a re-application of the original loading as it will to a reversal of the original loading. Unfortunately, as Polakowski points

out, the Bauschinger effect is found in tests on single crystals (40)

(22); possibly because of the existence of internal stresses within

the crystalline lattice structure. However, Polakowski found the idea of the existence of an internal stress system of value in the interpretation of the hardening and softening produced by plastic deformation.

He pointed out that when the material is deformed under the action of a. load system that differs from the one that set up the system of internal stress a new system of internal stress will be produced. For this to occur the old system of internal stress must be broken down.

Any breakdown involves a reduction in the existing stresses and this means that a transient softening of the material will occur. When the new load system begins to impose its own pattern of internal stress the material will begin to harden again. He states that the greater the difference between the two patterns of internal stress the greater the Bauschinger effect. Thus there is a large Bauschinger effect when a specimen is tested in tension after plastic deformation in compression and and a small Bauschinger effect when it is tested in tension after deformation by drawing or rolling.

For quantitative studies of the Bauschinger effect it is necessary to define some measurement which can be used to give a: numerici value to it. This can be done in a variety of ways. Broom and Ham (22) measured the plastic strain produced by a tensile stress and then reversed the loading and measured the plastic strain produced by a compressive stress of the same magnitude. They then divided the reversed strain by the original strain to give a quantity (-m.1)- the magnitude of which provided an inverse measure of the Bauschinger effect.

Studies in which proof stress changes are used as a measure of the

Bauschinger effect, as in the work of Dubuc (11), suffer from the defect that the results are very sensitive to small changes in the value of the strain specified for the measurement of the proof stresses.

From the point of view of the study of the behaviour of metals under cycles of plastic deformation the Bauschinger effect cannot be represented by a single quantity. The shape of the stresn-strain curves must be known. As a result of this almost all of the investigations that have been made - both on'poly-crystalline and single crystal specimens - do not provide information relevant to the present work. •—4.5-

3 DESCRIPTION OF APPARATUS AND SPECIMENS

3.1 General

The machine:available for the present work was an Instron universal

instrument, model TT-C, having a maximum load capacity of 10 000 lb.

The control system incorporates some features that are peculiar to

the individual machine (Series No. A/0019). The recording system

supplied with the machine was suitable for use with certain kinds of

axial extensometer. For the purpose of the present work it was nec-

essary to design a diametral extensometer and to modify the chart

drive system so as to increase the effective amplification.

3.2 The Testing Machine

3.2.1 Drive system. The machine is powered and controlled electrically. The

The drive systen,is designed to provide a_ constant velocity of crosshead

movement. The crossheadiscirivenby two four-start screws, of 1 in. pitch. These screws rotate in nuts attached to the moving crosshead. The mach-

ine has been modified by the manufacturers for use in cyclic tension-

compression testing. Two special anti-backlash nuts below the main

nuts can be adjusted by the user to eliminate any backlash between the

crosshead and screws. A modified thrust bearing at the top of each

screw eliminates any backlash between the screw and the fixed cross-

head of the machine.

A choice of crosshead speeds from 20 to 0.002 in/Min is provided;

but the very high speeds cannot be used for cyclic tests..

3.2.2 Load weighing system: The load cell, to which the upper end of the -6-

specimen is secured, is bolted to the fixed crosshead of the machine.

An electronic load-measuring system with bonded wire strain gauges

is used (!4.2). The load is recorded by the movement of the pen across

the chart 'of a Leeds and Northrup high-speed recorder.

A load selector switch allows the full-scale range to be changed

in steps from +200 lb to +10 000 lb. Calibration is carried out, using

dead weights, with the selector set to the 200 lb scale. As the switch

and its resistances can be checked separately this dead-weight calib-

ration applies to the whole range of load scales.

The recorder sensitivity is adjusted by means of an infinitely

variable control. Thus the chart can be made to read in units of

nominal stress if so desired. 3.2.3 Strain measuring system. Strain is recordedlythe vertical movement

of the chart beneath the pen. There are two alternative systems for

controlling the chart:

(1) The movement of the chart is made proportional to the move-

ment of the crosshead. This system is suitable for use with long cyl-

indrical specimens because crosshead displacement is then very nearly

proportional to strain; it is useless for waisted specimens.

(2) The movement of the chart is controlled by the electrical

signal from an extensometer mounted on the specimen. Various kinds

of extensometer element can be used. Only the system which uses a

microformer as a measuring element is described here.

A microformer is a small transformer the iron core of which is free to move in the axial direction. Movement of the core varies the

voltage output of the transformer. There is. a single primary winding

to which a constant alternating voltage is applied. There are two

secondary windings which are connected in series but with their output

voltages opposed to each other. When the core is in a central position

the two secondary voltages are equal and the net output is zero. When

the core is displaced from its mid-position there is a net voltage output.

Over most of the range of core travel the voltage output is sensibly

linear in relation 'to the displacement.

The system as supplied uses a pair "ofidentical microformers:

one is actuated by the lever system of the extensometer and the other

by a fine -screw, the rotation of which is proportional to that of the

chart drum. The secondaries of the two microformers are connected in

series with each other. .Zig... 2 shows:, in diagramatic form, the micro-

former and the basic circuit. If the cores of.the microformers are

not in identical positions, the individual picroformer outputs differ

and there is a net voltage output. This total voltage is passed to

the recorder amplifier which controls,the motor which drives the chart

drum. Thus the out-of-balance signal causes a movement of the chart

microformer core towards as position identical with that of the extenso-

meter microformer core. (A phase difference distinguishes between

the two directions in which a core may be displaced from its mid-

position.) Thus, throughout the test the movements• of the chart

-microformer core follow those of the extensometer microformer core and the vertical displacement of the chart is proportional to the strain

in the specimen.

Fig.* shows_ the actual circuit supplied. A potentiometer is con-

nected across the extensometer microformer secondaries. The purpose

of this is to provide an infinitely variable calibration control. By reducing the signal from the extensometer the potentiometer reduces

the chart displacement for a.given strain. A press-button switch

allows the potentiometer to be by-passed. When this is pressed the

chart microformer adjusts its voltage to zero, i.e. the core takes up

its mid-position.. It is ideal to operate the microformers with their

cores near their mid-positions because this gives equal travel in

both directions and avoids any possible non-linearity towards the

ends of the range.

3.3 The Specimen and End-Fittings

3.3.1 Choice of shape. Specimens having a long parallel portion have been

used in some of the earlier investigations into the behaviour of metals

under a. few reversals of stress (9)(18). They have the advantage that

they permit the use of an ordinary axial extensometer but they are

unsuitable for use in cyclic testing because buckling of the specimen

occur.% at quite small values of compressive strain. A waisted specimen

is much more resistant to buckling. A specimen having a parallel port- . ion at the waist of a. length just sufficient to take an axial-extenso-

meter with a short gauge length was used by Dubuc (11). He avoided the

necessity of extending lam parallel part beyond the gauge length by the -49-

use of a special fairing curve, which he claimed gave a uniform stress throughout the parallel part of the specimen. Such extensometers are complex in design and difficult to use. For accuracy it is essential that they should be of the type that gives an average reading of strain and thus eliminates the effect of any slight bending of the specimen.

If a diametral extensometer.is used the the portion of the spec-

imen actually under test can be very short indeed. Some workers prefer to retain a short parallel portion and to apply the diametral eaten-

someter to the mid-point of this. It is, however, simpler to omit the

parallel portion altogether; for if any tri-axiality of stress exists it will not be made smaller by the use of a short parallel portion. The easiest and cheapest shape to manufacture is one in which the profile consists of a- single circular arc and this was the form chosen for the specimens used in the present tests.'

The smaller the radius of the arc the greater the resistance of • the specimen to buckling but the larger theadius-of the arc the

nearer the approach to a state of uniform tensile or compressive stress across the test section. After some preliminary tests on EN25 steel a radius of 2 in. on a neck diameter of 1/14- in. was found to give a specimen that satisfactorily withstood buckling. These'dimensions gave a

Bridgman factor of 0.985 CO. Thus any error due to tri-axial stress

may be neglected. These dimensions were used for all specimens tested.

Aluminium specimens proved slightly more susceptible to buckling; but this was only a problem at large strain amplitudes. -50-

It is reasonable to suppose that geometrically similar specimens of a material will show a similar resistance to buckling . at a given stress- The larger the specimen the greater the accuracy with which given strain-

measuring equipment will be able to record the strain. The advantage

to be gained from an increase in waist diameter must be balanced against

the rise in the cost of the specimen; the materials cost, alone, will

vary as the cube of the diameter.

In the present work the maximum size of the specimens was lim-

ited by the capacity of the testing machine. A.neck diameter of IA

in. was used; this permitting the machine to apply a maximum stress of + 91 tons/in2.

Button-ended specimens similar in form to those used by Coffin

(3) were used. These are shown in Fig. 4. This design of specimen

has the disadvantage of having an appreciable length of unsupported

stem within each endfitting but it is simple in design and the locating

surfaces are easy to machine. When buckling did occur it was usually

localized in the waist of the specimen.

3.3.2 The end-fitting. An end-fitting is shown in part sectional view in Fig. 5 . A split collet locates the button-end laterally. A nut forces the collet against the button end of the specimen and thus

• presses the end of the specimen against the face of the end-fitting. The upper end-fitting screws onto the nose of the load cell

spindle and the lower and-ftting has a flange which is bolted to the

upper face of the moving crosshead. The screws that secure the flange -51-

permit a small amount of lateral displacement of the flange when the

specimen is being set up. A locating spigot was deliberately omitted when designing the flange. When the nuts are tightened friction supplies all the necessary restraining force. The load cell spindle is designed to have a fairly large resistance to lateral or angular deflection thus both ends of the specimen are mounted under conditions in which no angular or lateral movement is possible. These are the conditions most favourable for resistance to buckling.

Any kind of redundant restraint is, of course, a highly undesir-

able feature in the mounting of specimens for tests on materials. The ideal end condition is that provided by a frictionless ball joint. As this problem has never been satisfactorily solved in the case of tensile specimens it was felt that no useful purpose could be achieved by trying to design a ball joint fitting for tension-compression specimens.

3.3.3 Variation of stress across the test section. When load is applied to the specimen it is assumed that a state of uniform tensile or compressive stress will exist at the teat section. It follows from this that the specimen must be initially free from large-scale internal stresses. The existence of internal stresses on a granular or sub- granular scale is excluded because these stresses must be present except, possibly, in the case of annealed material - and their existence is an essential part of the problems of plastic behaviour which

are investigated in the present research. 52-

Consider an initially stress-free specimen that has been secured in place in the machine. If the specimen and machine are geometrically perfect the specimen can remain stress-free; but if any misalignment exists internal stresses will be set up in the specimen. The final connection is made by bolting the lower end-fitting flange to the surface of the crosshead. The flange is restrained laterally by the friction of the joint; so a small error in the lateral position of the flange will not cause internal stresses in the specimen. If the flange face is not parallel to the surface of the crosshead a bending moment must be applied to rotate it until it is parallel. It is possible by a simple test to find the value of the bending stress produced in the test section by any angular misalignment that may be present. The method is to bring the flange slowly into contact with the crosshead and (using the 200 lb load scale) to record the load at which complete surface-to-surface contact occurs. This point is marked by a sharp rise in the rate of increase of the load. This value of load, together with the radius of the flange face, enable the bending moment at the waist to be obtained. From this value the bending stress at the test section can be calculated. It was estimated that if the gap at the edge of the flange at the instant of first contact was

0.001in. the bending stresses would be; 1 ton/ins for aluminium,

11 ton/in2 for copper, and 3 ton/in2 for steel. Normally the gap was considerably less than this. Where a gap existed no attempt was made to pack it with shims; however carefully this is done some misalignment remains. The influence'of any initial stresses in the specimen on the shape of the stress-strain curve is always to cause a departure from the straight lime which represents elastic deformation. As a result low values of proof stress are observed. Fortunately, the effect of plastic deformation is to reduce the magnitude of the residual stresses and this effect occurs during successive cycles of plastic deformation.

Thus the internal stresses described above will be reduced to negligible values during the first few cycles of plastic deformation.

Lastly, consider a specimen which is known to have no internal stresses when mounted in the machine but which is not geometrically perfect. The applicatin of load to this may set up bending stresses in addition to the direct stress. Such stresses - caused, for example, by the centre of the test section being offset from the centre line of the endfittings - will not be eliminated by cyclic deformation of the specimen. They may may increase because of progressive buckling of the specimen.

The serious error that can be caused by a small eccentricity in the force applied to a tensile specimen was pointed out by Unwin in

1887 (43). When a specimen is tested using self-aligning shackles an eccentricity which is i',L of the specimen diameter results in a bending stress which is 8% of the mean stress. The effect of using a specimen with fixed ends will be to reduce this value but it will still be considerable. Examination of some specimens both before and during cyclic testing showed that, except after cycling at very large -54-

strain amplitudes, the eccentricity was always less than 1% of the neck diameter. It has been shown by Coffin (29) that, apart from buckling, a form

of plastic instability occurs during tension-compression testing which causes a decrease in the profile radius of the specimen and thus produces

an increase in any tri-axiality of stress that may be present. The greater the profile radius the smaller its rate of change; thus, providing a large profile radius is used, a very great number of cycles can be carried out before an appreciable change in shape occurs. The change

of profile ocouring in some of the aluminium specimens used in the present tests was investigated and was found to be negligible. 3.4 The Diametral Extensometer and Modified Chart Drive ystem 3.4.1 General. It was necessary to design a diametral extensometer which would use a microformer as a transducer and would operate with the exis-

ting chart drive system. It was to be suitable for strain amplitudes + + from about 10 down to 0.1% axial strain. A scale of 2 in. to 1%

axial strain was considered suitable. Axial extensometers. used with the Instron testing machine usually have a 2 in. gauge length and a lever system which magnifies the movement imparted to the microformer core. To give the same performance a diametral extensometer working on a neck diameter of 0.250 in. needs sixteen times as great a magnification. As the present work is concerned with plastic strains that are quite large compared to the elastic

strains it was decided that a smaller increase in magnification (x10) -55-

would suffice. Since the design of a mechanical system of magnification

presents considerable problems an attempt was made to provide the magnif-

ication by electrical means. Fortunately it was found possible, by some

simple modifications to the existing circuits to obtain a magnification

just sufficient for the purposes of the present research. 5.4.2 The extensometer. • The diametral extensometer is shown in Fig. 6 .

'It consists of two telescopic brass tubes through which are passed two

steel pins. One pin is located by one tube and the other by the other

tube. When fitting the pins care was taken to make their centre lines

parallel so that measurements would not be influenced by any slight

lateral movement of the extensometer in relation to the specimen during

a test. The body of the microformer is gripped by one end of the outer

tube and the core of the microformer is in contact with a screw which

projects from the flat end of the inner tube. A small spring inside

the microformer holds the core in contact with the end of the screw. A

spring4the lightest which would overcome the friction between the tubes)

is situated between the end of the microformer body and the end of the

inner tube. This spring exerts a force which tends to bring the pins

- together.

It has bean observed that in cyclic testing in the plastic range

quite a small. contact force will cause indentation of the specimen sur-

face tcr occur (7). A transverse steel pin, as used in the present work,

will, clearly, be less liable to cause indentation of the specimen

.than a knife-edge or a ball contact. In strain cycling any indentation -56-

of the contacts must result in an increase in the mean diameter of the specimen. Checks on the diameters. of several specimens after testing showed no evidence of appreciable indentation having ocaured.

It is essential that the extensometer should, throughout the test, contact the specimen at the teat section and that it should remain sensibly horizontal. In some preliminary testa it was mounted between two horizontal rods; one of which was carried by the upper end-fitting and the other by the lower end-fitting. A pair of suitably positioned rubber bands connected the exteneoneter to the upper rod and another pair, in similar positions, connected it to the lower rod. All were of equal stiffness and all were kept in tension. The object of the system was to ensure that, whatever the position of the end-fittings, the extensometer remained horizontal and level with the waist of the specimen; for, whatever the deformation, the waist should remain mid- way between the end-fittings. This suspension system was satisfactory in operation but it was cumbersome and hard to adjust.

A simpler system, in which the extensometer was supported at its mid-point by means of a spring bracket attached to the moving crosshead, as shown in Fig. 7, was then tried. This proved quite satisfactory provided that care was taken when setting up the extensometer to ensure that the spring bracket kept the pin carried by the body of the extensometer in contact with one side of the specimen.

Some error must occur when the extensometer tilts as the specimen deforms but comparative tests using the two systems showed no appreciable -57-

o error. A tilt of 1 would give an error of amial strain of about 0.03%. o To obtain a tilt of i it is necessary for the overall length of the

specimen to change by about 0.1 in.. This would only occur if the strain

in the test section was very large.

3.4.3 The modified circuit. Since no mechanical form of magnification was

to be provided in the extensometer it was necessary to arrange that a

small movement of the extensometer microformer core would cause a prop-

ortionately larger movement of the chart microformer core. In order

to achieve this the microformers sHould haveli linear relationship

between the output voltage and the core displacement; subsequent calib-

ration tests showed this to be true for almost the full range of travel

of the cores b Further it was necessary either to amplify the signal

from the extensometer microformer or to attenuate the signal from the

chart microformer. It was decided that the latter was the simpler

method as it could be effected by the use of a potentiometer; to achieve

the former an amplifier having linear characteristics would have to be

used.

Fig.3(0 shows the modifications made to the existing circuit: A is

the potentiometer, B and C are two adjustable resistances which may be

placed in series with the potentiometer. Thus the'selector switch makes

available a choice of three scales.

Attenuation of the signal from the chart microformer results in

a loss of driving power at the chart. For a given error of chart pos-

ition the available restoring torque at the chart win be reduced by the -5S-

same factor as that by which the microformer signal is reduced. It was

thought that it might be. necessary to provide an amplifier to increase

the torque at the chart drum; such an amplifier not needing to possess

linear characteristics. Fortunately there was found to be a large reserve

of power. The chart drive motor normally_ drives not only the chart

drum but, through friction clutches, the rolls of used and unused

paper. The drive to these was disconnected and a free length of paper,

weighted to keep it taut, hung over the chart drum. Under these con-

ditions very little torque was needed to rotate the drum. It was

found that the performance was further improved by using a trans-

former to step up the voltage of the signal received by the amplif-

ier. The existence of any backlash in the micrometer or in the train of gears that connect the recorder microformer core to the drum would res-

ult in a failure of the chart to follow exactly the movement of the

core whenever a reversal of motion occurs. Calibration tests show

this error to be very small. This was fortunate, because in order to

eliminate backlash it would be necessary to pre-load the drum to res-

ist its rotation in one direction and this would have greatly increased

friction in the system.

3.5 Calibration of the Scales 3.5.1 Calibration of the stress scale. The load scale was set to read in + terms of nominal stress. The following settings were used: 10 tons/in2

full scale deflection for aluminium, +25 tone/ina full scale deflection -59-

+ for copper, 100 tons/in2 full scale deflection for steel. For the highest setting the voltage across the load measuring potentiometer had to be raised above the normal value of one volt; but -'this had no adverse effect on the functioning of the recorder. Stresses in excess of 91 tons/le were not applied to the specimen as this would have meant exceeding the maximum load capacity of the machine.

The procedure when setting the recordeir strews scale_ was as follows: The zero load position of the pen was brought tO the centre of the scale. The load selector was set to the '200 ib' position and two 25 lb weights suspended from the load cell. The recorder sensitivity control was then adjusted until the pen was displaced 2.28 units from the zero position. In the case of steel 1.14 units had to be used.

Each unit is approximately 1 in.; there being 10 units to the total width of the chart. It was possible to carry out this setting to an accuracy of +1 part in 100.

The stress scale was normally calibrated once a day. This was done because gradual changes in the potentiometer battery voltage occur with time. In order to re-calibrate the specimen was released by dis- connecting the flange of the lower endfitting from the lower crosshead and the loads suspended from a rod which was passed through a hole in the upper endfitting. This could be done with the extensometer still in place on the specimen. Prior to calibrating the machine the electrical zero of the potentiometer is made to coincide with the zero on the load scales. It is -60-

convenient to do this because changes in the setting of the load sel•

ector switch do not then necessitate a re-setting of the point of zero

load on the scale. Also it permits a check to be made from time to time,

during a test if so desired, on whether any creep of the zero position

has occured. The pen should return to the correct zero posttion when

the button that short circuits the amplifier of the load measuring

system is pressed.

The accuracy of the load selector switch and its resistances was

checked by applying, a. load_and observing the change that occurred in

the displacement of the pen when the switch setting was altered. By

repeating this process, using larger loads for the larger ranges, the

ratio of all the settings could be checked.

To check that the calibration of the scales was the same in comp-

ression as in tension a test was made on the 10 000 lb load scale using

a compression proving ring.

Both these checks support the estimate of an accuracy of 1 part

in 100 of the full scale reading of the stress scales. This degree

of accuracy is quite acceptable in the present tests.

3.5.2 Calibration of the strain scale. The strain scale was calibrated

in terms of axial strain. It was assumed that no change of volume

occurs during plastic deformation and that the axial plastic str.in

is twice the diametral plastic strain. For strains of less than 5;1;

the error of this assumption is negligible.

Elastic strain does not occur at constant volume, so the elastic -61-

line on the stress strain graph produced will have a false slope. This does not matter if, as in the present work, the graphs are to be used for comparative purposei. If desired the graphs may be converted to a basis of plastic strain by deducting the 'apparent elastic strain' values. The following settings were used: 1 in. of chart movement represents i% (scale H); 1 in. of chart movement represents 1% (scale M); and

1 in. of chart movement represents 2% (scale L). To obtain these settings it was necessary that a relative movement between the tubes of

0.006 25 in. should cause chart displacements of 10.00, 5.00 and

2.50 in. for scales H, M and L respectively.

The displacement of 0.006 25 in. was used when adjusting potentiometer A and the resistances B and C prior to the preliminary calibration of the system. It was also used for the periodic checks that were made on the accuracy of the calibration.

In order to calibrate the extensometer it was placed in the clamp of a bench type micrometer. The pin was removed from the outer tube so that the micrometer spindle could contact the pin that is carried by the inner tube. Thus, while holding the outer tube, the micrometer could impose small and accurately measured displacements on the inner tube. The accuracy of the method of calibration was verified by the use of another method; in which Johansahon slip gauges were inserted between the pins of the extensometer. This gave good agreement with the former method. The use of the bench micrometer proved the most -62-

convenient method and, indeed, was the only satisfactory method when

it was desired to investigate the possible presence of errors due to backlash. The micrometer drum was marked with 0.0001 in. divisions and the settings were estimated to be accurate to the nearest tenth

of a division.

A step by step calibration of each strain scale was carried out

to verify the linearity of the relationship between the strain and

chart displacement and also to ascertain whether backlash or friction

effects were present. At the start of the calibration the micrometer

was adjusted to bring the microformer cores to their mid-positions.

The extensometer microformer core was then displaced first in one

direction and then in the other. For scale H the full range of chart

movement possible (about 30 in.) was used. Fig. 8 shows the three

calibration graphs as they appeared when plotted on the chart itself.

The full range of movement is shown only for scale H. If scale H gives a linear relationship for the full range then the other scales

will also give a linear relationship.

Only for strains ofmore than 6% do the points for the calibration

of scale H depart from the straight line-. Even at 72 the deviation

is only 3 parts in 100 of the recorded value. The results of the

three calibrations, as determined from the slopes of the lines, show

errors of 0, 0 and 3 pertain 100 for scales H, M and L respectively.

The results shown in Fig. 8 are for a re-calibration after a period of testing. Normally gfter adjustment of the potentiometel there was -63-

no detectable error in the slope of the graph.

The points in Fig. 8 show a minute backlash or friction effect.

At its worst it is about 0.05% strain and it is usually a lot less.

This estimate is supported by the results shown in Fig.9 which illus-

trates the performance of the recording system when used with a work-

hardened copper specimen under conditions in which unloading, reloading

and reversals of load occur. The Bauschinger effect and various forms

of hysteresis loop are illustrated. Some of the very narrow loops

that occur under conditions in which no plastic deformation is expected

may be attributable to anelastic effects; but the presence of backlash

and friction effects can also cause loops. The smallest anelastic

effect will be expected in the cases when a small reduction of load

is followed by its re-application. Where such conditions approaching

a state of perfect elasticity occur Fig. 9 supports the evidence

provided by the calibration test, that backlash and friction effects

are very small.

A reasonable estimate of the accuracy of strain measurement with

the modified recorder system would seem to be that it is better than + 1 part in 100 of the largest strain recorded on each scale.

6 The Strain Cycling Controls

6.1 General. The Instron testing machine, as supplied, was provided

with controls for automatic cycling between preset limits of either

load or crosshead position. The former system was used in the present

work as a means of control between nominal stress limits; but the

:;o : 9, 11, 17. 41,52. 53 and 54 are replicas of autographic record. _6t,.-

latter could not be used for strain cycling because with waisted spec-

imens the strain at the test section bears no fixed relationship to the

overall length of the specimen. It was therefore decided to modify

the existing crosshead cycling control system so that strain, as

measured by the diametral extensometer, would control the reversals of

movement of the crosshead.'

3.6.2 Modifications to the crosshead cycling control system. In the system

as supplied the drive screws. are connected to two adjustable cams.

Each cam, on reaching a selected position, operates a single pole two-

throw micro-switch. Operation of the switch reverses the direction of

motion of the crosshead. One cam controls the upper and the other the

lower limit of crosshead movement. The form of control circuit used

need notbe2onsidered here because the only changes that have been made

have been the replacement of the reversing mipro-switches by relays.

Since chart displacement is proportional to strain switching dev-

ices actuated by the chart drive system are needed if cyclic tests are

to be made using pre-selected limits of strain. Insterid of designing

a ,systemofmechanically operated switches for attachment to the chart

drive system it was decided to try a scheme in which conducting strips

of metal foil attached to the face of the chart acti:ate the existing

control system. After such a system had been tried and found successful

it was retained because it was realised that it could be used for

limits of plastic strain as well as for limits of total strain.

The additional equipment provided is shown, in diagramatic form, -65-

in Fig. 10 . Two narrow strips of aluminium foil are stuck to the

surface of the chart. Two spring contacts are associated with each

strip: One is a fixed "contact attached to the chart p.la ten; this

makes contact first and then slides over the vertical portion of the

strip. The other is a moving contact attached to the pen carriage;

when this makes contact with the strip it closes the circuit which

operates one of the reversing relays.

Reversal of the chart drum does. not occur immediately the direc-

tion of motion of crosshead movement is reversed. Some creep effects

are present - as can be seen in the curved portion that occurs at the

commencement of each unloading line in Fig. 9 . In order that the pen

point shall not reach the metal foil the two pen contacts are set at a

distance of 1/4. in. from the pen, one above and the other below it.

When setting up the apparatus for given plastic strain limits the

strips are offset by 1/4 in. less the estimated over-run of the chart.

3.6.3 Plastic strain limit control. The use of moving contacts mounted on

the pen carriage instead of a simpler system of fixed contacts has the.

adv,ntage that limits of plastic strain can be used instead of limits

of total strain. Tf the foil is placed horizontallyq across the chart

reversals will occur at selected limits. of total strain; but if the

foil is placed parallel to the lines which represent elastic deformation

then reversals will occur at selected values of plastic strain.

Fig. 11 shows the stress-strain curves. for a.cold-worked copper

specimen tested at a plastic strain amplitude of +1%. A crosshead -66- speed of 0.005 in/Min was used; this giving one cycle in about 45 seconds. The strain control system does not function satisfactorily at higher speeds. The use of a constant crosshead speed does not ensure a constant strain rate at the test section; the strain rate increases as the limit is approached and the final value can be much higher than the average for the cycle. It is the final speed of approach of the pen to the strip which limits the crosshead speed that may be used. The crosshead speed of 0.05 in/Min was adopted for use in the present work; except for a few tests carried out at specially high or low speeds to investigate the effects of strain 41. rate.

Under favourable conditions the strain limits can be set and + held to an accuracy of 0.05;0 plastic strain. If the material is very susceptible to creep the limit cannot be set so accurately.

If the top part of the curve suffers a considerable change of slope during cycling the limit cannot be so accurately held. -67-

4 MAThaaLS AND RESULTS 4.1 Copper

4.1.i Condition as supplied. The copper was suppled in the form of two

10 ft lengths of 3 in. diameter bar made from certified lengths of

oxygen free high conductivity (OFHC) copper. These bars had been

annealed by the manufacturer at 650° for two hours and lightly reeled.

No analysis was available.

The average Vickers diamond pyramid hardness number obtained

from specimens made from the bar was 60. It was found that re-

annealing an individual specimen could reduce the hardness number I x to 47. This reduction in hardness is small compdred to the changes in

hardness being investigated - the most severely cold worked copper

had a hardness of about 120 - and it was, decided not to anneal the

individual specimens. The'heat treatment of individual specimens,

particularly if done in batches in small furnaces, is liable to increase

the variation in hardness from specimen to specimen.

4.1.2 Annealed specimens. These were all cut from the 3 in. diameter bar as shown in Fig. 12 . In the case of copper the term 'annealed'

refers to the 'as supplied' state of the material. Thedimensions

of the specimens are shown in Fig. 4.

4.1.3 Stretched specimens. Some specimens, having the normal profile

and neck diameter, were machined 'from the 3 in. diameter bar and then stretched in simple tension by various amounts prior to cyclic testing.

These were intended only for preliminary investigations into the -68-

behaviour of cold worked material. It was not found convenient to

re-calibrate the machine for each new neck diameter. However, the

observations on the specimen that had recieved a reduction of area

of were included in the recorded test results, a small correction

being applied to allow for the reduced diameter of the neck.

Other specimens, having the normal profile but an oversize neck

diameter, were machined from the 3 in. diameter bar and then stretched

in simple tension until micrometer measurements at the waists showed

that the required reductions of area had been achieved. The original.

diameters were chosen so that the necks were still over-size after

stretching. The specimens were finally machined to give the normal

profile and neck di ureter.

Specimens str,!.(a. to give 10, 15, 20 and 502,; reductions of

area were produced. One having a reduction of 45o appeared satisfac-

tory but failed, slightly below the waist, when a cyclic test was

commenced

4.1.4 Swaged specimens. Cylindrical bars 7/B in. diameter and 4 in.

long were machined from the 3 in. diameter bar. The centre portions

of these were then press swaged by various amounts to give the required

reductiota of area. The bars were then machined to form specimens

having the normal dimensions. The swaging was done using an improvised

set of dies in a compression testing machine. Although intended orig-

inally for use in preliminary investigations it was found that, in

spite of the crudeness of the method employed, these specimens ,G-

gave results suitable for inclusion with those obtained from the

other specimens. An 80/0 reduction of area was achieved for one spec-

imen, this being the most'heavily cold worked specimen dealt with in

this investigation.

4.1.5 Cold drawn specimens. 30 in. long bars were machined from the 3 in.

diameter copper bar as shown in Fig. 13. The diameters selected were

such as would give reductions of area of: 10, 20, 30, 40, 50, 60 and

70 when each was reduced to a final diameter of 0.375 in.. This

diameter gave a bar of just sufficient size for the production of

the specimens. The supplierscould not draw material from diameters

larger than 1.23 in. and the two largest bars recieved their initial

reductions by a cold swaging process. The dies used had diameters of:

1.195, 1.156, 1.151, 1.097, 1.065, 1.053, 1.007, 0.982, 0.950, 0.912,

0.898 and 0.875 in.. Thus the amount of reduction per pass varied,

being, on the average, slightly more than 5, reduction of area. The

three bars that had suffered reductions of 50, 60 and 70% all proved

defective when tested,. In some cases flaws could be observed with

the naked eye near the axis of each bar. It has been suggested that

these defects arose in consequence of the use of many light reductions

in the drawing sequence. This had been unavoidable because of the

limited capacity of the drawbench available.

Thus the specimens produced by the process of drawing had recieved

reductions of area of 10, 20 30 and 40%.

4.1.6 Roller swaged specimens. 15 in. long bars were machined from the in. diameter copper bar as shown in rig. 14. The diameters selected

w ere such as would give reductions of area of: a0, 50, 60 and 70, C.''75 when each was reduced to a final diameter of in.. These bars

were roller swaged from alternate ends using .a reduction of area of

12"/0 at each stage (except for the final stage). Thede four bars

were. like the drawn bars, all reduced to a final diameter of 0.875 in..

They all proved quite satisfactory during test.

Thus the specimens produced by the process of roller swaging had

aecei,ed reductions of 40, 50, 60 and 70.

!,.2 Aluminium

4.2.1 Condition as suaplied. The aluminium was supplied in the form of

in. diameter bar. The bar had been extruded from a billet of commer-

cially pure aluminium to specification 2SM. The analysis of the material

is given in Table 1.

The average Vickers diamond pyramid number obtained from specimens

made from the bar was 32. It was found that-annealing individual spec-

imens at 425°C reduced their hardness number to 25.

Although there was little scatter in the hardness numbers of the

annealed specimens it was found that extremely erratic results were

obtained -when cyclic tension-compression tests were carried out on

them. It was clear that the bar, as supplied, was slightly work hard-

ened; due, perhaps, to extrusion at a rather low temperature. Now it

is known that material that has undergone a very small amount of cold

work is very susceptible to grain growth during annealing and that the -71-

size of the grains is very sensitive to small changes in the amount of

cold work. (See Fig.6 and 7 of (44).) The hardness of the bar as

0/b and it supplied corresponds to a reduction of area of• less than 2 is therefore probable that large grains grow during annealing and that

there may be a wide variation in the grain sizes produced in a number

of specimens. Attempts to check the grain sizes by etching a number

of specimens did not, unfortunately, provide conclusive evidence.

To ensure a uniform size of grain in the annealed specimens it

is necessary that, prior to annealing, the material should have under-

gone a reduction by cold working of at least 20.. The suppliers

suggested that they should reduce the diameter of the bar to 2.312 in.

by drawing, this corresponding to a reduction of area of 40/-L The

average Vickers diamond pyramid hardness number obtained from specimens

made from the drawn bar was 41.

No attempt was made to anneal the bar after drawing because the

production of the specimens from the fully soft material would have

presented grave practical difficulties.

4.2.2 Annealed specimens. These were cut from the 2.312 in. diameter bar

as shown in Fig. 15. The dimensions of the specimens are shown in

Fig. 4 . The specimens were annealed in small batches after manuf-

acture. Unless otherwise stated a temperature of 355°C for 11 hours

was used. The average value for the Vickers diamond pyramid hardness

number obtained from the annealed specimens was 21.

4.2.3 Cold drawn specimens. Some specimens were left in the cold drawn -72-

state as described above; that is to say in a 'half hard' condition.

4-5_En25 Steel

4.3.1 Condition as supplied. The En?5 steel was supplied in the form

of 1 in. diameter bar which had been annealed at 655°C for 2 hours, o cooled at 10 C/bour to 630 C and then air cooled. The analysis of

the material is given in Table 2. The cast had a grain size of 5

4.3.2 Annealed specimens. The/annealed specimens were machined from the

1 in. diameter bar. The dimensions of the specimens are shown in

Fig. 4 . The average Vickers diamond pyramid hardness number obtained

from.specimens made from the bar was 274.

4.3.3 Hardened and tempered specimens. Specimens machined from the 1 in.

diameter bar were hardened by soaking at 830-850°C and quenching in

oil. The average hardness number for specimens in tie hardened state

.was 514.7. The specimens were then tempered at various temperatures.

After holding at each temperature for z hour the specimens were allowed to cool in the_furnace.

During heat treatment the specimens were wired together in small

batches and placed in a box of lime. The use of small batches caused

some variation in the heat treatment conditions from specimen to spec-

imen. However, by the use of suitable tempering temperatures a select-

ion of specimens of various degrees of hardness from very nearly dead

hard to almost fully annealed was produced. • 4.4 Stainless Steel

4.4.1 Condition as supplied. The stainless steel was supplied in the form -73-

of 2 3/4 in. diameter bar in the fully soft condition. The analysis of the material is given in Table 2.

4.4.2 Specimens. These were all cut from the 2 3/4 in. diameter bar aa shown-in Fig. 16. The dimensions of the specimens are shown in Fig. 4 . 4.5 Results for Tests on Copper 4.5.1 Cyclic tests between fixed limits of plastic strain. Fig.17 shows the as-recorded stress-strain loops for an annealed copper specimen tested at a plastic strain amplitude of +0. The loops for a cold- worked copper specimen subjected to the same strain amplitude are shown in Fig. 11. Although the cyclic hardening or softening shown by these specimens was considerable it will be noticed that the change per cycle decreased rapidly. Autographic records similar to Figs.11 and 17 have been used to . prepare the various graphs in which the results of this investigation

are presented. Both tension and compression values are plotted,.aince

in some cases they are appreciably different"ef. Fig..49a Alas**

otherwise stated the first stress applied in eyalie testing OILS allays tensile. A few comparative tests showed that- sitilarresultS ate obtained when the first stress applied is compressive. Fig.18 shows the results of testing annealed copper specimens at various plastic strain amplitudes. At strain amplitudes between 0.5 and 1% there is a definite indication that the stress tends towards a limiting value for each strain. range. Tests at higher strain ranges

are inconclusive because the specimens failed. The onset of failure -74,-

was usually shown by the development of a difference between the nominal

compressive and tensile stresses - a fine crack in a specimen reduces

the tensile load it will support but has little effect on the compressive

load because the crack closes under compression. Tests at lower strain

• amplitudes are inconclusive because prolonged tests were not possible.

Control is difficult at small strain amplitudes and sometimes tests

had to be terminated earlier than had been intended.

Where more than one specimen had been tested at a particular strain

amplitude the results for all were used to prepare the curve. An exam-

ination of the scatter of the points for curves produced from pairs of

specimens , cf. curves +0.46 and +1.49 of Fig. 1.8- , shows that' different

specimens give very nearly the same values of stress and that after the

first few cycles the values agree closely.

The results for the first and second cycles of a. test occasionally

gave points which lay appreciably above or below the curve. Because

of this a broken line is used for the first portion of each curve. It

is probable that such variations arose from an initial non-uniformity

of stress across the test section; this being eliminated by a few cycles

of plastic deformation. Preliminary hardness tests made on the specimens

suggested that there was likelY to be little variation in the mechanical

properties from specimen to specimen.

Fig. 19 shows the results of tests at various plastic strain amplitudes for work-hardened copper. It will be seen that when the strain

amplitude is small the difference in tensile Etn8 comOressive properties -75-

persists for a number of cycles. For the strain amplitudes investigated it was found that the 40 and 70% cold-worked material always softened whereas the 10A cold-worked material hardened and/Or softened according to the strain amplitude.

Fig. 20 shows the influence of various methods of cold-working on the cyclic hardening of softening of copper cycled at a plastic strain amplitude of 1%. Comparison between the various methods of cold-work = roller swaging, press swiaginT,,,drawing and stretching - is only possible in a few cases but the results suggest that the method

Of cold-working has little or no effect on cyclic hardening or softening.

Fig. 21 shows the influence of the degree of cold-working on the behaviour of copper cycled at a. plastic strain amplitude of +1,-c;. Olen the initial plastic deformation by cold work was 20% reduction of area' or more the material always softened. When it was 15% or less an initial hardening of the material occured and, except in the case of the annealed material, this was followed by softening. Fig. 21 provides some support for the view (28) that stress amplitude tends to a limiting value for each strain range irrespective of the prior strain history. Fig. 22 shows the results presented in Fig.18 plotted on a base of cumulative plastic strain instead of the number of cycles; the cumulative plastic strain being four times the plastic strain amplitude multiplied by the number of cycles. These curves are shown as chain lines (without points) and the inotonic stress-strain curve .(T) has been added for the purpose of comparison. This form of presentation shows that, for a given cumulative plastic strain, the greater the number of cycles the less the degree of hardness produced in the annealed material. It also shows the rapid levelling off of the curves; which, because a logarithmic base is used in Fie. is not readily appreciated when that graph is examined.

To investigate further the effect of prior strain history each of a set of annealed copper specimens was tested at a series of strain amplitudes. At each strain amplitude cycling was continued until the stress amplitude tended to a limiting value. Thus a specimen when cycled at one strain amplitude may be considered to have an initial state of hardness produced by a history of cycling at one or more lower

(or higher) strain amplitudes. The results for these step-tests are plotted in Fig. ;'2 (full lines) and it can be seen that they support the view that the limiting value of stress amplitude for a given strain amplitude is independent of prior strain history.

Fig. 23 shows the cyclic stress stIein curves for annealed copper and for copper that has been roller swaged to 70:':,• reduction of area. In these curves stress amplitude is plotted against strain amplitude for given numbers of cycles. The data is present in Fig:,. 18 and 19 but in that form it cannot readily be used in design calculations. In the cyclic stress strain curves it is presented in a form convenient for designers. Although the graph is on a base of plastic strain the curves can be read in terms of total strain - which is what designers -77-

need - by the addition of an inclined line, representing the elastic

stress-strain relationship, to the left of the vertical axis. The

simple tension stress-strain curve for the annealed and cold-worked

material are included with the cyclic stress-strain curves; they are

the curves for the first quarter cycle.

Fig. 24 compares the results obtained from two annealed copper

specimens both of which were tested at a plastic strain amplitude of

+1T the initial loading being tensile in the case of one and compressive

in the case of the other. The curve is a reproduction of the one in

Fig.lb .

4.5.2 Cyclic tests between fixed limits of nominal stress. If it is true

that the limiting value of stress amplitude for a given strain amplit-

ude is independent of prior strain history then a material cycled

between stress limits should tend to the same settled cyclic state

as a material cycled between the corresponding limits of strain. Tests

were made on a single specimen of annealed copper cycled between nom- + inal stress limits of 6.2 tons/in2. After the initial deformation (of

about 2% strain) little change in the mean diameter of.the specimen was

observed. The cyclic hardening of the material showed itself in a

gradual narrowing of the stress-strain loop. The strain amplitude

was found to be: +0.09% for the 2nd cycle, 0.06% for the 50th cycle + and 0.06: for the 100th cycle. With so small a strain amplitude it

was not possible to make accurate measurements but the amplitude seems

to tend towards a limiting value of 0.06%. In the tests made between fixed limits of plastic strain neither the specimen tested at 4-0.05% + nor that tested at 0.07% achieved a settled'Oyclic state. However, the general trend of the family of curves suggests that the settled cyclic stress amplitude corresponding to a strain amplitude of 0.06% + is between 5 and 7 tons/in2. Thus the results obtained by stress cycling are not very different from those obtained from strain cycling. + For a decisive test a nominal stress amplitude of, at- least, 10 tons/in2 would have to be used. This is not practicable because of the large changes in the mean diameter of the specimen - the first loading would produce a strain of about 10

An annealed copper specimen was then tested at a series of stress • amplitudes; these tests corresponding to the step-tests used in strain cycling as described in the previous section. By this means quite large stress amplitudes could be achieved without a large change in the mean diameter of the specimen occuring. given so, at the larger amplitudes the tests were complicated by the occurrence of cyclic creep.

Each cycle did not give a closed loop in the stress-strain diagram and the 'loop width' had to be taken as the mean of two values. However, following step-tests at seven lesser amplitudes, 50 cycles were carried + 2 out at a nominal stress amplitude of 10 tona/in . • The material appeared to have reached a settled cyclic state and the plastic strain amplitude was observed to be Despite the use of progressive changes of stress amplitude there had been an appreciable reduction in the mean diameter of the specimen by the completion of 50 cycles at -79-

+ the nominal stress amplitude of 10 tons/in2. The exact amount by

which the cross-sectional area of the specimen had altered is not known because it had been necessary to re-set the extensometer during the

tests. It was estimated to be less than 10% of the original area.

A direct measurement could not be made because it was desired that

the extensometer should be left in place to enable further tests to be made. Thus the true stress amplitude used -was slightly less than + 11 tons/in2 The results for tests made between fixed limits of

plastic strain show that a specimenitested at + 0.46% strain tended to a

settled stress amplitude of +10.2 tons/in2. Therefore there is fairly close agreement between the results for the tlAo kinds of cycling.

In the specimen of annealed copper that was cycled between nominal

+ • stress limits of tons/1n2 slight cyclic creep occurred. It has

been pointed out by Benham (31) that with the the tensile and compressive

stresss- limits set to the same numerical values cycling will result in a gradual elongation of the specimen. He observed that the rate of this cyclic creep is very sensitive to small changes in the stress limits.

With the apparatus used in the present work it is easy to make small changes in the mean Stress while leaving the stress range unchanged.

This is done by altering the position of the axis of zero stress on

the chart of the autographic recorder.

Fig. 25 shows the results of a series of tests made on a specimen

using a constant range of nominal stresz of tons/In2. In the first test the mean nominal stress was zero and the cyclic creep -80-

was only just measurable. In further tests on the same specimen the mean nominal stress was given progressively increasing tensile values.

At each fresh setting it was observed that the first loading caused a relatively large tensile deformation but that after about 20 cycles the material appeared to reach a stable state. The creep per cycle was then measured. In Fig. 2 the creep strain per cycle is plotted against the value of the mean nominal stress. It would appear that a very small compressive mean nominal stress would suffice to stop or reverse the cyclic creep.

The mean nominal stress, was next changed to zero - i.e. the specimen was, for a second time, cycled between noMinal stress limits of + .25tons/in 2. It was expeeted that the specimen would, after a few cycles, show the same settled cyclic behaviour that had originally been observed for this stress range. The intention was to show that no apprecaible change of material properties had occurred during the intervening series of cyclic tests. 30 cycles were carried out between these stress limits and the result was not what was expected. Instead a small cyclic deformation accompanied by a barely measurable cyclic creep it was found that after the first cycle the material, to all . appearances, was perfectly elastic.

At first sight this behaviour appeared to be in complete disagree- ment with the results obtained from the step-tests described in the previous section. These indicated that cycling between fixed limits of plastic strain - however small the deformation - should cause the -81-

material to soften or harden towards the appropriatelstable7zyclic state. If a material possesses an elastic range and if it is cycled between stress limits that lie within this range then it will not suffer -cyclic plastic deformation. It seems improbable that elastic deformation - or anelastic deformation - alone can cause a'ohange in the mechanical properties of a material. Thus it would seem that the reason why the specimen behaves elastically when cycled between nominal + stress limits of 6.25 tons/in2 is that the previous test had hardened the material sufficiently to ensure that - after the first cycle at

+6.25 tons/in2 had lowered the elastic limit in tension and raised it in compression - the new elastic limits exceeded the limits of the applied stress. This phenomena was further investigated using a. specimen made from copper that had been cold worked by roller swaging, to) 70% reduction of area. This was found to yield in tension at s: stress of about 24 tons/In2, while - because of the Bauschinger eRfectits elastic lit in- compression was very low.- .The value:for thisioan only be,„eptimstedL but the reversed loading curve of Fig.30auggests that itt,isAletwaen 0' and 6 tons/in2. If the first cycle" van produce. symultryonfctensiIs, and compressive elastic limits without altering the elastic range then cycling between nominal stress limits of 12.0 tons/in2 should not (after the first cycle) cause plastic deformation of the material. +4 The specimen was cycled between nominal stress limits 2.5 + 4.0 + and 7.0 tons/in2 without appreciable departure from elastic behaviour. -82-

It was next subjected to 1000 cycles between nominal stress limits

of 7.6 tons/in2 and appeared to be fully elastic. Nor did 1000 cycles + between nominal stress limits of 12.6 tons/in2 reveal any appreciable

plastic deformation. However, when the nominal stress limits were

raised to 15.2 tons/in2 a very small cyclic plastic deformation was

observed. As the cycling continued the width of the stress-strain

loop increased, showing that a softening of the metal was occurring

A very small amount of cyclic creep occurred but this did not interfere

with the conduct of the test.

Fig. 26 shows the plastic strain range plotted againinst the

number of cycles. The last part of the curve is shown in broken line

because it is believed that a fatigue crack began to form in the spec-

imen at about 3000 cydles. It will be observed that not only does

cyclicg at 15.2 tons/Ina cause the material to soften but as the

plastic strain range increases, the rate of softening increases. H&ci

'failure not occurred thehardness must have.eventually tended,towards

a settled cyclic state (a plastic strain amplitude of about.+0:404.

It could not have continued decreasing indefinitely.

Some tests were made on specimens cycled between zero stress and

a maximum tensile stress. To avoid the difficulties presented by the

large initial deformations that occur when an annealed specimen is

tested work hardened specimens were used for these tests. A trial

test using a nominal stress limit showed that the cyclic creep that

occurs in each cycle is very sensitive to the slightest variation in -83-

stress range. It was realised that, while satisfactory for tests in which the stress limits are of opposite sign, the use of nominal stress: limits for tests in which the stress limits remain wholly tensile (or compressive) could give a false picture of th material's behaviour. A progressive reduction in specimen diameter causes a progressive rise in the values of the true stress limits and this may result in a rapid increase in the rate of.cyclic creep.

The study of cyclic creep requires large numbers of stress reversals and,..because of the precision and reliability of the load limit controls of the machine, cyclic tests between nominal stress _Limits can be made with'accuracy and ease. Thus, despite their disadvantages, nominal stress limits were used in the present work. However, as a check on the phenomena described above, two tests were attempted using true stress limits.

it is possible to use the strain cycling controls in conjunction with the metal foil strip for cycling between true_stress limits. To do this the foil must be attached to the chart at a small angle to the strain axis. (In strain cycling it is at a small angle to the stress axis.) The angle was s.11ected so that the load limit varied in direct proportion to the cross sectional area of the specimen. Thus the true stress limit was held constant. For a zero stress limit (which called for no correction) it was found convenient to use the load limit control.

Tests were carried out on specimens made from copper that had been cold worked to /4-0 reduction of area by drawing. This was found to yield in tension at a stress of about 20 tons/Ina . That is to say it suffered -91,—

a large elongation when this stress was reached; though the first plastic deformation occured at an appreciably smaller stress. One specimen was tested between true stress limits of 0 and +19.2 •tons/in2. A small plastic deformation (about 0.1/u) oocurred on the first application of load and over the next 50 cycles a barely perceptible cyclic creep was.h observed.

A second specimen was then tested between true stress limits of 0 and +20 tons/in2. The initial loading caused a plastic deformation of about

0.5/:: strain. It then appeared that the specimen had settled down to a cyclic creep rate of about 0.12; per cycle. Some irregularity in the creep from cycle to cycle was expected; because it is highly sensitive to small errors in the stress limits. The variation was not great except that at intervals of 7 to 13 cycles a considerable elongation - as much as 0.8 plastic strain - was observed. After such sudden exten- sions the steady rate of cyclic creep was' resumed. It is possible that this curious phenomena represents the behaviour of the material. Any irregularity in the stress limits would be expected to cause a random variation in the cyclic deformation, not an intermittent effect.

During one cycle the speed of the machine was changed to C.24_ times normal rate. This did not affect the rate of cyclic creep. Nor did similar checks during other cyclic tests on copper show any time effect.

This test suggests that when cycling between limits of true stress progressive increase or decrease in the creep rate does not occur. Its occurrence incyclic tests made at wholly tensile (or compressive) stresses must he due to geometrical effects. -85-

Some further tests were carried out on specimens made from copper that had been cold-worked to 206 reduction of area by drawing. This was found to yield in tension at a stress of about 17 tons/in2. The object of these tests was to ascertain to what extent cyclic creep occurs when the range of stress is too small to cause cyclic softening of the material. One specimen was tested between nominal stress limits of +10.7 and -5.3 tons/in2 - a stress range chosen because it should provide conditions very favourable to cyclic creep. The range of stress being slightly less than the probable elastic range of—the material it was expected that no cyclic softening would occur. During 1000 cycles no measurable softening was observed and the cyclic creep was found to be very small indeed. This specimen was then further tested for + 1000 cycles between nominal stress limits of 8.2 tons/62 - a: stress range slightly greater than the previous one - and neither softening nor cyclic creep was observed. The specimen was then tested between nominal stress limits of 0 and 16.5 tons/in2 - the same stress range as in the last test. A plastic deformation of 1.1'io occurred on the first application of load. Cyclic creep then occurred at•a rate of about 0.12% percycle. After about 20 cycles the cyclic creep.began to increase rapidly and soon became so large that the test had to be stopped. Another specimen was tested between limits of nominal stress tlit were both tensile, the.upper limit being kept at +16.5 tons/in2 while the lower limit was given a series of values. In the first test the limits wore +12.2 and + 16.5 tons/in2. A small deformation -86-

(corresponding to a plastic strain of about 0.21 occurred on the first

application of load. A small cyclic creep was then observed and this

was found to decrease during the first few cycles and then to have a

rate of about 0.002./,, per cycle. Similar tests were then made using

lower stress limits of +10.0, +7.3, +2..-.3 and +2.7 tons/in2. The

rate of cyclic creep rose slightly from test to test to a value of

about 0.00/4b in the last test, but in no case was cyclic softening

(as shown by a widening of the stress-strain loop) evident. Finally

the specimen was tested between nominal stress limits of 0 and +16.5

tons/ins. The cyclic creep was at first observed to be about 0.02%

per cycle but, because of the progressive reduction in the specimen

diameter, the rate of creep soon began to increase and excessive defor-

mation made it necessary to end the test after 100 cycles.

In order to compare cyclic creep in compression with cyclic

creep in tension tests were made on two specimens made from copper

that had been cold-worked to 50% reduction of area by roller swaging

This was found to yield in tension at a stress of about 22 tons/Jr?.

In compression it was_ found that plastic deformation first occurred at

very small stresses. One specimen was cycled in tension between

nominal stress limits of 0 to + 17.7 tons/in2 for 200 cycles, then

0 to +20.2 tons/ins for 200 cycles and then 0 to +20.6 tonsile for 200 cycles. In each test, apart from some very small plastic deformation on the first application of load, the material showed a barely

perceptible cyclic creep and there was no sign of cyclic softening. It was then cycled between nominal stress limits of 0 to +21 tons/le" for 450 cycles. At first the cyclic creep was only just perceptible, but it increased from cycle to cycle and by 450 cycles had become So large that the test had to be stopped. It would appear that had this unstable behaviour been avoided - by the use of true stress limits instead of nominal stress limits - a. stable and very small cyclic creep would have been found. The other specimen was cycled in compression between nominal stress limits of 0 to -11.2 tons/in' for 200 cycles ,then 0 to - 15.3 tons/io2 for 200 cycles, then 0 to -.16.1- tons/in'for 200 cycles and then 0 to -17.7 tons/ins to 200 cycles.

In each test, apart from some plastic deformation on the first application.of load, the material showed a barely perceptible cyclic creep and there was no sign of cyclic softening. It was then cycled between nominal stress limits of 0 to -21.0 tons/in'. The first loading produced a plastic deformation of about 0.857; and cyclic creep was then observed to occur. Contrary to what would be expected from the results of previous tests there was a gradual increase in the creep from:cycle to cycle. This must indicate a change in the materiel's properties fo'r the increase in the cro'is'seCtional area of the specimen during cyclic creep causes a progressive decrease in the upper limit of true stress and this should result in an apparent decrease in the creep rate from cycle to cycle.

The tests on these two specimens suggest that for the same limits. of nominal stress a considerably greater rate of creep occurs when the -88--

stresses are compressive than when they are tensile. That this is

so may well be due to the asymmetry of mechanical properties produced

when the material was roller swaged.

These last tests reveal very clearly that, even when compression

is used, satisfactory information about changes in material properties

during cycling between zero and a maximum nominal stress limits is

only obtainable when the test is•restricted to small total strains. 4.5.3 Other tests. Fig. 27 shows the stress-strain curves, as produced

by the autographic recorder, for annealed copper tested in tension and

in compression. In the case of tension the specimen was tested to the

point where the load began to decrease; and in the case of compresion

it was tested to the point where buckling occurred. For the larger

strains it was necessary to change the strain scales to the M and L

settings. Where this was done the missing portions of the curves

were plotted using the values from the other curves.

For strains of up to 10.5` true stress values were plotted forboth

tension and compression and the points were found to lie on a single

curve, thus showing that the mechanical properties of the copper in

tension and compression were sensibly the same.

Fig. 28 shows the stress-strain curves for annealed copper loaded

in compression after deformation in tension. Each curve was obtained

from an individual specimen. To this diagram have been added similar

Ourves (cholla in broken line)- obtained from tests in which single

specimens were subjected to increasing cycles of plastic deformation. -89-

Fig. 29 shows the stress strain curves for one of the specimens. The reversed loading portions of these were reproduced in Fig.._28 in the positions appropriate to the maximum tensile stress that had been applied. There is (with one exception) fair agreement between the curves obtained from individual specimens and those obtained from the repeated testing of single specimens. Thus there is some evidence that when the stress last applied exceeds all previous stresses the

Bauschinger effect - as represented by the form of the curve for reversed loading - depends solely on the magnitude of the last stress applied before the reversal.

The agreement in shape of the curves is not so close as to justify the labour of re-plotting this diagram using true stress values instead p of nominal stress values. The correction factor differs for the two types of curve; thUs at large strains the broken line curves are in- correctly positioned on the strain axis. If, however, the reversed loading curves differ in- scale, but not in geometrical form, over an appreciable range of strain then a, satisfactory comparison can be made using nominal stress values. When the present research was commenced it was not intended to make a detailed study of the Bauschinger effect in annealed materials.

It was realised that any small variation from specimenrtospaoimen, . due, perhaps, to internal.stresses-presint,at:tha;start,Of.Itesting,...- that might reveal itself in the stress-strain turve,forthe first cycle would be eliminated during the first few cycles. Thus a number of the -90-

specimens used for cyclic testing ha' small abnormalities in their stress-

strain curves for the first application of load. Only those that confor-

med closely to the tensile stress-strain curve of Fig. 27 were used to

give the full line curves of Fig. 28.

Pig. 30 shows the stress-strain curves in tension, as produced by

the autographic recorder, for copper having various initial states of cold work. Curves for a reversal of loading after a plastic strain of

10 are also shown. For ,4,0 reduction of area curves for both the drawn and the roller swaged material are shown (the former in broken line).

Curves for annealed and for stretched copper (broken line) have been included to give a more complete picture of the material's behaviour.

Annealed copper does not possess an appreciable elastic range but copper which has hardened to a sett]ed cyclic state characteristic of a particular strain (or stress range) does possess an ,elastic range.

This elastic range reveals itself in the stress-strain diagram a straight unloading line continued by a short line representing elastic behaviour diiring the initial stages of application of reversed loading.

It has been demonstrated, during the stress cycling tests, that when the elastic range is unsymmetrical about zero stress it can be rendered symmetrical by the application of a single cycle of deformation using

equal tension and compression limits chosen to give a stress range only slightly less than the elastic range. If, however, a sequence cf cycles of progressively decreasing plastic strain amplitude is applied a softening of the copper results and this softening must be accompanied by a -91 -

decrease in the elastic range of the material. Progressive cycling down towards a very small strain amplitude can apparently do two things:

It can soften the material; and it can increase the degree of symmetry between its behaviour in tension and in compression. Some tests were made in an attempt to ascertain whether these two effects could be separated.

Fig. 31 shows a set of stable cyclic loops obtained from a specimen of annealed copper cycled at +2.0% plastic strain for 15 cycles, +1.5% for 5 oycles, +1.0% for 10 cycles, +0.5% for 100 cycles, +0.2j. for 100 cycles and +0.06% for 100 cycles. As strain cycling between smaller limits was not practicable progressively decreasing cycles of stress were then applied to zero stress. A tensile test (to 2% strain) was then carried out (shown by the full line). This revealed that the material, though very much softened, was still a great deal harder than annealed copper (shown by the chain line). It appeared that the copper retained an elastic range and that this was about 5 tons/in2.

It is probable that had time permitted a far larger number of cycles to be made at the lower stress ranges a lower curve might have been produced.

The same specimen was then cycled at +2.0,4 plastic strain until it again reached a stable cyclic state (the loop for which conformed closely to the original one). It was again cycled down, but this time relatively few cycles

appreciable' softening 'had ()occurred during these few cycles of plastic

deformation, though much less softening than pravils

Figs. 32 (c) and (d) show.(in full line) the corresponding curves

for another specimen of annealed copper which had been subjected to

an initial plastic strain range of 4-0.5. In (e) many cycles have

caused a considerable softening of the material but in,(d) the few cycles have caused little softening though they_have reduced the degree

of asymmetry between the the properties of the material in tension .and compression. The extent of this asymmetry is shown in Figs. 32,_(4.), and (b). In each of these tests the material was cycled until the stable state was reached. The cycling was then interrupted se that on unloading the specimen was at the point of zero strain iL tna ease of (a) the unloading was done from the compression part of- the curve and in the case of (b) from the tension part of the ohzva. In bot cases a tension test was then carried out. The curves from (u) and

(b) have been superimposed (in broken line) on (c) and (d).

It will be seen that in Fig. 32 (d) the curve for thp matcrial that had. recieved a few cycles of progressively decreasing, straln lin between the curves that represent the full asymmetry of mechanical properties. However, so few cycles (less than 10) have ueen used in the cycling down process that it is doubtful whetl'er the curve is a unique one.

It is clear that copper. with its large capacity for cyciib soft- er.ing is not the best material to use to investigate this last point. -93-

If the stress-strain curve for reversed loading is determined by

the v,Ilue of the previously applied maximum stress and, subject to certain limitations, is uninfluenced by past strain history, then

it is reasonable to expect that the rate of strain hardening just

prior to the reversal of load should also depend sclely on the maximum applied stress.

Fig. 33 shows the results of tests made on two specimens of + annealed copper cycled between plastic strain limits of one for

5 cycles and the other for 25 cycles. In each the specimen was tested in tension beyond the cyclic range during the final cycle. Portions of the true stress-strain curve for annealed copper tested in tension

(from Fig. 27. ) for the corresponding levels of maximum stress have been superimposed on the diagrams. In each case the two curves can be seen to hate a similar slope; though, because of the low rate of strain hardening, the test is not very decisive.

Fig. 34. shows the results for a series of comparative tests made on specimens of annealed and.cold worked copper. One specimen of each pair was loaded in simple tension to a particular stress level followed by reversal into compression. The other was strain cycled to the same stress level as the former. The relevant parts of the stress-strain corves have been plotted for comparison. It will be seen that there is close agreement between the shapes of the reversed loading curves both for conditions of cyclic softening as well as for cyclic hardening and that there is agreement even when many cycles have been applied. 4.6 Results for Tests on Aluminium

4.6.1 Cyclic tests between fixed limits of plastic strain. As in the

case of copper autographic records were used to prepare the graphs

in which the results of this investigation are presented.

Fig. 35 shows the results of testing annealed aluminium specimens

at various plastic strain amplitudes. At strain amplitudes of 0.52X,

or less there is definite evidence that the stress tends towards a

limiting value for each strain range. The results for annealed

aluminium, though similar to those for annealed copper, show a lower

rate of cyclic hardening.

Fig. 36 shows the results of testing cold drawn aluminium specimens

at two different strain'amplitudes. In both cases softening of the

material occured; though, in the case of the specimen cycled between + plastic strain limits of _ 1.02/0 this was followed by an apparent hardening (shown in broken line). Although this apparent hardening was

undoubtedly a result of the geometrical instability effect described

by Coffin (7) an examination of the profile of the specimen showed

general buckling to have occured and this masked any change of profile

radius that may have been present.

Fig.37 shows the cyclic stress-strain curves for annealed alumin-

ium. Insufficient results were available to permit the corresponding

curves for cold-drawn aluminium to be plotted.

4.6.2 Cyclic tests between fixed limits of nominal. stress. Tests were

made on a specimen of annealed aluminium cycled'between nominal stress -95-

+ limits of 1.9 tons/ins. After the initial deformation (of about

0.594. strain) a very gradual cyclic creep was observed. No significant

hardening or softening of. the material was observed; the strain ampl- + itude remaining nearly constant at 0.06% for 1000 cycles. The cyclic

creep varied in an erratic manner and its average value over 1000 cycles was 0.002% per cycle. A second specimen of annealed aluminium + was cycled between nominal stress limits of 3.05 tons/in2. After

the initial deformation (of about 2.5) appreciable cyclic creep was

observed. At first this was about 0.18,0 per cycle and it gradually

decreased, over 50 cycles, to about 0.12% per cycle. A slight cyclic

hardening was observed; the strain amplitude decreasing from +0.3270 + to 0.28 during 50 cycles.

Fig.36 shows the results of tests made on the specimen that had + been cycled between nominal stress limits of 1.9 tons/in'. In each test the stress range was kept constant at 3.8 tons/In? while the mean nominal stress was given a series of small values, first in tension and then in compression. In Fig. 38 the creep strain per cycle is plotted against the mean nominal stress. As with copper, a small compressive mean nominal stress is needed if the cyclic creep is to be zero.

A specimen of cold drawn aluminium was cycled between equal limits of nominal tensile and compressive stress chosen to give a stress range slightly less than the elastic range of the material. This was found to yield in tension at a stress of about 6.2 tonS/in2 and was expected to -96-

have a low elastic limit in compression. Thus the elastic range might

be expected to be greater than 6 tons/in2. This specimen was cycled + betweeh nominal stress limits of 3.20 tons/in2 for 6700 cycles. A

very narrow stress-strain loop developed at the first cycle and the

strain amplitude - about +0.04% - remained constant throughout the

test. There was a minute amount of cyclic creep and the rate of this

did not appear to be increasing.

The erratic results obtained from specimens made from aluminium

in the 'as supplied' condition had suggested that the behaviour of the

material was very sensitive to chance differences in the structure

produced by annealing. The resulits obtained from material annealed

from the cold-drawn condition were far more consistent but not so

consistent as those for annealed copper. To check whether this was

due to variations in the annealing process a single specimen was anneal-

ed at 400C for 1 hour instead of the 355°C for 17 hours used for all

the tests described above. When the stress strain curves for 500

cycles at a plastic strain amplitude of +1% were compared there was

no appreciable difference between them.

A very light spring was used in the extensometer for all the tests

described above after it had been found that slight indentation by the

contacts was occuring when the original spring - found_quite satisfac-

tory when used with copper - was used.

4.6.3 Other tests. Fig. 39 shows the stress-strain curves for annealed

aluminium tested in compression after deformation in tension. Each -97-

curve was obtained from an individual specimen.

Fig. 40 shows the stress-strain curves in tension, as produced by the autographic recorder, for aluminium in both the annealed and cold-

drawn state. Curves for a reversal of loading after a plastic strain

of 1% are also shown. 4.7 Results for Tests on En25 Steel 4.7.1 Cyclic tests between fixed limits of plastic strain. Fig. 4.1 shows

the as-recorded stress-strain loops for a hardened and tempered En25

steel specimen tested at a plastic strain amplitude of 1%). Although

the cyclic softening shown by the specimen was considerable it will be

noticed that the change per cycle decreased rapidly.

Autographic records similar to that shown in Fig. 41 have been

used to prepare the various graphs in which the results of the inves-

tigations are presented.

Fig.42 shows the results of testing both annealed En25 steel

and En25 steel.Which.had been hardened, from 840°C, and tempered, at

500°C, at various plastic strain amplitudes. It will be seen that

the hardened and tempered steel cyclically softens at all strain ampl-

itudes. The annealed steel only softens significantly when the strain + amplitude is less• than

Fig.43compares the results obtained from two annealed En25

steel specimens both of which were tested at a plastic strain ampl- + itude of 1%), the initial loading being tensile in the case of one

and compressive in the case of the other. The ,tress-strain loops -98-

for the 1st and the. 80th cycles are shown. The signs have been reversed in the loops for the second specimen so that the pairs of.100pS_ are superimposed and their shapes can be compared. There is fairly close agreement between the cyclic behaviour of the material regardless of the sign of the initial loading. Although 1% strain does not suffice to deform the material beyond the extent of the yield step during the first application of load yet there is no sign of any yield step during the reversal of loading which follows.

Fig. 41+ shows the influence of the degree of initial hardness produced by heat treatment on the behaviour of En25 steel cycled at a plastic strain amplitude of +1%. Although the phenomenon of cyclic softening is as pronounced as with work hardened copper the first few cycles seem to play a smaller part in the_softening process in the case of steel.

Fig. 45 shows the cyclic stress-strain curves for annealed En25 steel and En25 steel that had been hardened, from 840°C, and tempered, o at 500 C. These curves show more clearly than does Fig. 4.2 the fact that significant softening of annealed En25 steel only occurs when the yield step is not exceeded during the, first application of load.

Fig. 46 shows the results of testing annealed En25 steel which had been work hardened by stretching to 10% reduction of area at a plastic strain amplitude of +170. Cyclic softening occured and symmetry of stress limits in tension and compression was achieved in about twelve cycles. In this instance the initial asymmetry is quite great. -99-

As with tests on other materials a check was made on whether a reduction in the rate of cycling caused any appreciable change in the shape of the stress-strain loop. No such change was found to occur.

However, it was observed that if a cyclic test had been interrupted for several hours and then testing was resumed the stress-strain curve for the first loading deviated from the corresponding curve in the last loop. After the first loading (the first 1/4 cycle) the curves conformed closely to the previous loop.

Fig. 47 shows the results of tests made to investigate the effect of different periods of rest on the reloading curve. The tests were made on two specimens of En25 steel that had been hardened from 840°C and tempered to 450°C and then cycled at a plastic strain amplitude of + until they reached a settled cyclic state. Periods of cycling

- during which the loop did not vary appreciably from its original shape - were alternated with periods of rest. 3 hours rest had an appreciable effect'on the shape of the re-loading curve and 66 hours rest produced a much greater effect. An extension of the rest period from 66 hours to 22 days Only caused a small change in the curve. The change in shape always resulted from'a re-appearance and rise of the (tensile),elastia.limit. and there was no sign of the re-appearance of the yield step. In the case of one specimen a rest period of 171 hours was allowed after the application of the first (tensile) load.

The reverse load curve was then compared with the one for the specimen that had hot:had a rest period. No significant difference could be seen. , -100-

4.7.2 Cyclic tests; between fixed limits of nominal stress. Tests were made

on a specimen of En25 steel that had been hardened, from 840°C, and

tempered, at 610°C. In this condition it was found to yield in tension

at 66 tons/in2 and - assuming a similar behaviour in compression - to have

an elastic range of about 130 tons/ins. This specimen was first cycled + + at 30 tons/Ina for 50 cycles, then at 40 tons/ins for 50 cycles and

then at "50 tons/in' for 50 cycles. It showed no signs of departure

from fully elastic behaviour. It was then cycled at 60 tons/in (a

stress range of 120 tons/in') for 20 cycles. The plastic deformation

in the first cycle was not measurable but it could be clearly seen in

the second cycle. It rapidly increased; giving a plastic strain range

of +0.u5,; in the 10th cycle and 0.12;:e in the 20th cycle. No appreciable

cyclic creep was observed.

No attempt was made to test a specimen using a cyclic stress, range

in excess of the elastic range because it was clear that - because of

the yield step and the low rate of work hardening - very large uni-

directional deformations would occur. 4.7.3 Other tests. Fig. 48 shows the stress-strain curves, as produced by the autographic recorder, for annealed En25 steel tested in tension and

in compression. In the case of tension the specimen was tested to the

point where the load began to decrease; and in'the case of compression

it aas tested to the point where buckling occurred„ For. the larger strains

it xas necessary to change the scales to the M and L settings. .Where

this was done the missing portions of the curves were plotted using the,

values from the other curves. -101--

For strains of up to 12.5 the true stress values were plotted

for both tension and compression and the points were found to lie on

a single curve, thus showing that the mechanical properties of the

•annealed En25 steel in tension and compression were sensibly. the same.

Fig. 49 shows the stress-strain curves for annealed En25 steel

loaded in compression after deformation in tension. Each curve was

obtained from an individual specimen. It will be noticed that in

contrast to the behaviour of copper and aluminium, the unloading line

is not straight and plastic deformation commences almost as soon as

reduction of the load begins.

Fig. 50 shows the tensile and compressive stresi-strain curves

for En25 steel hardened, at 840°0, and. tempered, from 550°C, together

with similar curves for annealed En25 steel. For:the purpose of pomp-

ari son the compression curve has been plotted with positive values for

both stress and strain. The values for true stress have been plotted

for the hardened steel and are found to lie very nearly on a single

curve. Thus, while the agreement is not so close as for the annealed

material, the mechanical properties of the hardened and tempered En25

steel are very nearly the same in tension and compression.

Fig. 51 shows the stress-strain curves in tension, as produced

by the autographic recorder, for En25 steel having various initial

states of hardness. Curves for a reversal of loading after a plastic

strain of 1% are also shown. These curves suggest that the non-linearity

of the unloading curve is less pronounced in the harder specimens. -102-

Advantage was taken of the small capacity for strain hardening possessed by En25 steel to make a thorough investigation of the phenomenon of elimination of asymmetry of.tensile and compressive properties by the application of a series of progressively decreasing cycles of strain.

The tests were conducted on a specimen made from annealed En25 steel.

This was first cycled between plastic strain limits of 1% for 100 cycles. The stress-strain loop had by then become stable. Each of the tests described below was followed by cycling at +1A plastic strain for sufficient cycles to re-establish the settled cyclic loop.

Fig. 52 shows the as-recorded stress strain curves for the following four tests: (1) The cycle was interrupted during compression so that on unloading the specimen was at the point of zero strain; it was then test- i ed in tension. (2) The cycle was interrupted during compression so that on unloading the specimen was at the point of zero strain; it was then tested in compression. (3) The cycle was interrupted during tension so that on unloading the specimen was at the point of zero strain; it was then tested in tension. (4) The cycle was interrupted during tension so that on unloading the specimen was at the point of zero strain; it was then tested in compression.

Fig. 53 shows the as-recorded stress-strain curves for the specimen when it was subjected to a series of progressively ;decreasing cycles of strain (about 30)_followed by a tensile test.

'Fig. 54 shows the as-recorded stress-strain curves for the specimen when it was subjected to a series of progressively decreasing cycles of -103-

strain (about 30) followed by a compression test.

In Fig. 55 the three tension and three compression curves from

the previous three figures are combined in one diagram. The curves for tests after cycling down (full lines) are identical in shape; thus

showing that symmetry of tension and compression behaviour had been achieved- Also these curves lie between the curves (broken line) that represent the maximum degrees of asymmetry. Thus symmetry was achieved

without a general softening of the material having occured; in the tests done on copRer this was not so.

;i:xamination of Figs. 53 and 54 reveals that the tensile (or compression) test curves pass very nearly through the peaks of the stress- strain loops. The curves so produced are therefore very nearly the shape of the limiting curve to which the cyclic stress-strain curves tend for a large number of cycles.

The fact that annealed En25 steel shows some plastic deformation after quite a small reduction of load suggested that it might be very

susceptible to cyclic creep when tested under conditions where the mean stress was not very much smaller than the maximum stress. Although stress cycling tests were not made this point was investigated. A specimen of annealed En25 steel was loaded to a tensile stress of 52 tonsAn2 and the stress then reduced to 40 tonstin2. No plastic deformation was observed during the unloading. tie-application of the stress of .52 tons/In2 produced a plastic strain of about 0.37i. The specimen was then unloaded to 30 tons/in2 - again with no visible plastic deformation. ite-applicatiOn of the stress of 52 tons/in2 produced a

plastic strain of about 0.25;L. The specimen was then unloaded to 20

tons/in2 - during which slight plastic deformation was obseived. Re-

application of the stress of 52 tons/in2 produced a plastic strain of

about 0.51,, - not a significant increase. Unloading to 10 and to 0

tons/in2 resulted in small plastic deformations and. the plastic strains

on re-loading were 0.23 and respectively. Even the reversal of

load and application of a compressive stress of 3 tons/in2 resulted in

wplastic strain of only 0.203/4 on re-application of the load. It was

clear that the variations in the plastic strain for re-loading were

random effects - caused, no doubt by small variations in the upper

stress limit. There was certainly no large increase in the plastic

deformation for re-loading as the lower stress limit was reduced.

A number of tests were made, using the same specimen, in an

attempt to discover the extent of the elastic range (if any) during

unloading in a cycle of plastic deformation carried out between fixed

limits of tensile and compressive stress. The unloading part of a

cycle was interrupted by two reversals of the direction of straining.

Generally this produced a loop within a loop on the stress strain

diagram. Only when the stress range of the inner loop was made very

small did the loop narrow towards a straight line.

4.8 Results- of Tests on Stainless Steel

4.8.1 Cyclic tests between fixed limits of plastic strain. As in the case

of copper autographic records were used to prepare the graphs in which -105-

the results of this investigation are presented.

Fig. 56 shmNs the results of testing stainless steel in the fully soft condition at various plastic strain amplitudes. The results for stainless steel show an initial hardening similar to that observed in copper - though occuring at a lesser rate. However, for strain amplitudes of 1'0.42/ or more the initial hardening was followed by a far more rapid hardening. The higher the strain amplitude the earlier the transition point occured and it seems probable that the specimens cycled at 0.25,'4 and +0.01'='; would also have shown an increased rate of hardening if the tests could have been prolonged to sufficiently large numbers of cycles.

Fig. 57 shows the cyclic stress-strain curves fon. stainless steel in the fully soft condition.

It was felt that the complications introduced by the change in the rate of cyclic hardening might make an investigation of the behaviour of the material under step-test conditions meaningless. However, a single specimen of stainless steel was tested at a series of strain amplitudes. 1000 cycles at +0.10X gave a settled stress amplitude of

16 tons/In2. 20 cycles at +0.03% gave a settled stress amplitude of

14 tons/in2. 20 cycles at +0.08% gave a settled stress amplitude of

17 tons/in2. 20 cycles at +0.12% gave a settled stress amplitude of

18 tons/in2. 20 cycles at +0.26% gave a settled stress amplitude of

22 tons/in2. 20 cycles at +0.51N gave .a settled stress amplitude of + 24 tons/ins 20 cycles at 1.1070 showed the stress amplitude to be -106-

30 tons/in". and.- as would be expected - still increasing. 10 cycles

at +1,98 showed the stress applitude to_be 41 tons/in' and-still increasing. The increase continued rapidly to 46 tons/in' at 20 cycles

- the transition point having been passed. Finally the specimen was

cycled at +0.934. plastic strain. After 20 cycles the stress range had settled down to 47 tons/in' and after 60 cycles it had only changed

to 50 tons/in'. Thus there is fair agreement with the settled values

of cyclic stress amplitude for individual specimens (see Fig. 56) for

strain amplitudes of 40.5 or less so long as the transition point is

not passed.

4.8.2 Cyclic tests between fixed limits of nominal stress. Tests were made

on a specimen of stainless steel cycled between fixed limits of nominal

stress of +20 tons/in'. The stress-strain loop rapidly approached a

stable state and at the 10th cycle the strain amplitude was +0.170;

while at the 500th cycle it had only reached +0.217,;. It was clear

that at this stress range a transition point could not be expected

until many more cycles had been applied. Another specimen was tested

at 25 tons/in'. The first. loading caused an initial strain ,of about 9.35% to occur,butafter this there was little change in the mean diam-

eter of the specimen. There was A'rapid increase in hardness during

the first few cycles.. A'slight decreaSe then occnred and it then

remained almost-copstant up to 100 cycles - the plastic strain ,amplit-

ude then being...0.03%. It then gradually increased up to 200 cycles -

the plastic strain amplitude then being +0.34%. -107-

Other tests. Fig.56 shows the.stress-strain curves, as.produced by the autographic recorder, for stainless steel in the fully soft condition in tension and in compression. In the case of tension the

specimen was tested to about 30 strain; at which the load was still increasing. In compression it was tested to only 6.5/0.strain. For

the larger strains it ws necessary to change the strain scales to

the M'and L settings. Where this was done the missing portions of

the curves were plotted using the values from the other curves.

For strains of up to 6.5o true stress values were plotted for both tension and compression and the points were found to lie on a

single curve. thus showing that the mechanical properties of stainless

steel in tension and compression were very nearly the same.

Fig.59, shows the stress-strain curves for stainless steel in the fully soft state loaded in compression after deformation in tension.

::.b.ch curve was obtained from an individual specimen. It will be noticed that, as with En25 steel, the unloading line i; not straight and plastic deformation commences almost as soon as reduction of the load begins.

,The curves for stainless steel in the fully soft condition showed that the material had no yield step and that the elastic limit is very low.

Time did not permit an investigation of the influence of time - as against number of cycles - on the position of the transition point.

In view of a possible time factor tests were conducted without any breaks. -108-

5 DISCUSSION 5.1 Survey of Cyclic Tets Between Fixed Limits of Plastic Strain

5.1.1 Effect of various initial states on the yclic behaviour. In the

case of both copper and aluminium cyclic hardening of the annealed

material was observed to occur at all amplitudes of plastic strain;

the increase, for a particular number of cycles, being large for large

amplitudes and appearing to tend to- sero for very small amplitudes.

The tests on both materials support the view that,in the cases of

topper and_aluminium the stress amplitude tends towards a limiting

value for each strain amplitude. This general behaviour is in agreement

with that observed by Ludwik and Scheu (19) and subsequent workers in

this field.

The rate of cyclic hardening was found to be much greater in the

case of copper than in the case of aluminium despite the fact that

these metals - which both have a face-centred cubic structure - have

tensile stress-strain curves of similar form.

Stainless steel in the fully soft state initially showed a cyclic

hardening behaviour not unlike that of copper and aluminium; though

oecuring at a much slower rate. That the rate should be lower is not

unexpected because the tensile stress-strain curve for stainless steel

shows a low capacity for strain hardening.

Annealed En25 steel, when tested at strain amplitudes sufficiently

great to exceed the yield step on the first loading, showed no signif-

leapt changc in the hardness of the material. The tensile stress-strain -109

curve for annealed F,n25 steel showed the material to have little capacity for strain hardening so little cyclic hardening could be

expected. It is, however, surprising that the slight trend that can

be detected in the curve of Fig. 42 suggests that, after the first few cycles gradual softening might commence. It is possible that the material - which is difficult to anneal - was not in its softest possible condition; though preliminary hardness tests suggested that it was fully soft.

In the case of cold worked copper hardening and/Or softening was observed to occur according to the strain amplitude. (Fig. 194 If the, initial hardening was high in relation to the strain amplitudesoftening occured; if it was low hardening occured. In the intermediate range softening followed an initial hardening. For particular degrees.of initial hardness the rate of softening did not vary greatly when the strain amplitude was changed; except at small amplitudes - at which the value of the strain amplitude determined whether initial hardening or initial softening occured. Cyclic softening was also observed in the small number of tests made on work hardened aluminium and work hardened En25 steel. (Figs. 36 and 46.)

Tests on copper cold worked to similar reductions of area by various methods show that the method used has little or no effect on the cyclic hardening or softening process. (Fig. 20.)

For a particular strain_amplitude (4.1q annealed copper had a high rate of initial hardening but copper having progressively increasing -110-

degrees of cold-work showed decreasing -rates of initial hardening until at a particular value of hardness (corresponding to about 18% reduction of area) no initial change occured; thereafter increases in degrees of hardness produced increases in the rate of initial softening. (Fig. 21.)

If the curve for annealed copper in Fig. 21 tends asymptotically to the horizontal line representing a particular stress level then it is inconconceivable that the curves for the cold worked copper should desbend below this line. There is some evidence (in'Fig. 21 ) that oopper'having various initial degrees of hardness tends towards the same limiting stress amplitude when cycled at a particular strain amplitude. This supports the view, expressed by Polakowski (6), that there is a stable cyclic state corresponding to each strain range.

The results for the one specimen of strain hardened En25 steel that was tested are inconclusive because both the annealed and cold worked materials were still softening when the tests were concluded. Also the tests on cold worked aluminium had -to be discontinued while the material was still softening. HoweVer, there is nothing in the results for En25 steel or aluminium that ill contrary to the, idea: that a.etable cyclic state;exists. (See Figs. 36 and 44.) The various methods of cold working used in the present research all render the material anisotropic. Thus the properties of the cold worked material differ in tension and compression; the difference being most marked for small deformations. This difference represents the Bauschinger effect. The degree of asymmetry of properties differs -111-

slightly with the._ method of cold work (38); the extremecase beipg the cold working of the material by.stretching. The evidence of all the strain cycling tests made on cold worked materials is that the lack of symmetry present in the first cycle is progressively reduced during subsequent cycles. (A very clear illustration of this, for cold worked En25 steel is to.be seen in Fig.4.6 .) When the strain amplitude is large in .relation to the degree of cold work the restor- r. ation of symmetry occurs in a few cycles; but when it is small many cycles are required. A symmetrical stress-strain loop does not mean that the material properties in tension and compression are symmetrical and this point is discussed in 5.3.1 below. The importance of this initial lack of symmetry in the interpretation of the results of the strain cycling tests is that there is not a single curve for each strain amplitude but a pair of curves that converge to form one. Only in cases where the initial difference was large were both branches of the curve indicated. (cf the small strain amplitude curves of Fig.19.) Thus, though the use of stress amplitude as a measure of hardness is of great convenience in cyclic

strain testing under certain conditions it is unsatisfactory. It is not possible to measure hardness - which however it be defined is, surely, a non-directional property - by means of a quantity which has, one of two values according to the choice made between tension and compression. Fortunately, in the present tests, the initial difference was usually small and soon decreased; being often masked by the -112-

variation in the observed values of stress. Had the present investigation been extended to smaller strain amplitude:this phenomenen: would have proved very inconvenient. It should be noted that Polakowski and Palchouduri (2 ) - although they applied very-small-cyclic.; strains and used a compression test as a measure of hardness - did not encounter this difficulty. In stopping their cyclic tests they were forced to carry out a series of cycles of progressively decreasing amplitude and, in so doing, they rendered the properties of the material symmetrical in tension and compression. This was also the case with the work of several other investigators.

The introduction of asymmetry of the stress-strain loop by initial cold working can be avoided if an entirely different kind of deformation is used for the cold working to that used in the cyclic testing. This has been done by various investigators - often as a matter of convenience. It could have been done in the present investigation by using thin walled cycindrical specimens and hardening them by torsional deformation. However, it would appear that in a fundamental investigation into cyclic hardening an softening the same kind of strain system should be used both for initial hardening and for the cyclic teats. Though this was not practicable for the whole test programme the principle was adhered to as far as possible.. Where it had to be abandoned - in the use of swaging, drawing, etc,-in place of stretching in simple tension - the comparative tests (described above) shOwed that consistent results were being obtained. . -113-

The existence of a settled cyclic state dependent on strain amplitude

but independent of the initial state of cold work established by

Coffin and Tavernelli (28) and supported by the results of the present

investigation suggests that - at least in the case of copper and aluminium - the settled cyclic state for a given strain.amplitude

might be entirely independent of previous strain history.

The ability of cyclic plastic deformation to soften a previously

work hardened material raises the question as to whether a similar

softening occurs in materials that have been initially hardened by

heat treatment. In 1962 Mackenzie and Benham (45) reported some

observations made on heat treated En25 steel during an investigation

of the low endurance fatigue behaviour of that material then in progress

at Imperial College. They showed that, with the steel hardened and

tempered to give a tensile strength of 70 tons/in2 , considerable

softening of the material occured when it was cycled at strain amplit- + + udes ranging from 0.67 to 4.8):; and that, after sufA.cient cycles, the

material tended towards a limiting state of hardness.

The results of the present investigations, in which En25 steel

having a wide range of states of initial hardness was tested, confirm

Mackenzie and Benham's results - though an even moref rapid initial

softening was observed - and show that the phenomenon of cyclic

softening is even more striking when the steel has a greater degree of of initial hardness. (Fig•. 44.) The' present tests were not, however, continued to a very large number of cycles and therefore do not provide evidenca of the: existence of .a settle& .cyclic' state.

Bairstow had shown (17) that cyclic plastic deformation can develop when a steel is cycled at a plastic strain amplitude that

does not exceed the yield stress. Conversely, it is therefore to be

expected that when a eleel le cycled at asTlastio strain amplitude

which does not exceed the yield step strain the stress amplitude

will decrease. The-effect 'ofe-Such-cyclingAs.to eliminate the yield

step phenomenon. Thus for strain amplitudes which do not exceed the yield step strain very different behaviour is be expected compared

to the behaviour 'Art- the larger strainamplitudes. This reveals itself

in the results of the tests made on annealed En25 steel - which

shows only slight changes in hardness when cycled at plastic strain amplitudes in excess of 1;.?;. (Fig. 42.)

Thus the marked softening of En25 steel at small strain amplitudes

is an entirely different process to the'tycIiC'softening'or.hardening

that ocours in:metals in. general. It is presumably due to a progressive unlocking of the dislocations which, in the original state are stopped from moving by the presence of interstitial carbon atoms (1+6). For large. strain aMp4tudes.thefirst defofmation Suffices to.unlock all the dislocations but with the small cyclic strains discussed above the unlocking process will be a progressive one, revealing itself in a progressive decrease in the stress amplitude. Although the, yield_

step 'is no longer to be seen,after- the first application of load - even for the case of very small strain ranges - this does not mean -115-

that all the locked dislocations have been freed during the initial I. loading.

Of all the metals tested in the present investigation the only

one to show a ehminge in the form of the stress-strain, loop after ;a

period of rest was En25 steel. This change, (Fig. /47.) Thioh consisted of...

a reappearance of the elastic line -,but without any- re,-mppearance of

the yield step - at the first loading of the resumed cycling must be associated with the time dependence of the degree of reaction between

solute atoms and dislocations.

It is known that in repeated tension tests a time delay can

result in a re-appearance of the yield step and this is believed to

be caused by a migration of the solute atoms (4hl. Bauschinger's

tests showed that the effect of a time delay was much more gradual with the reverse loading curve than with the re-loading curve. Yet;

in the present tests - where continuation of cycling begins with a

reverse loading curve - some time effect could be detected after only

a few hours delay.

The yield step phenomena is a very complex one; so much so that an attempt to explain its influence on cyclic behaviour is likely -

to achieve little. It is known that yielding does not proceed uniformly

throughout the material under test. Thus it may well take an entirely different course in a waisted specimen, as used in the present tests, as against a cyclindrical specimen , in which an appreciable mass of

material is under test. Further, the progress of the development of -11L-

the zones of material that has yielded must cause the stress distribution

in the test section to be far from uniform. Only when the settled

cyclic state has been achieved can it be said that the behaviour of

the specimen is really decided by the cyclic properties of the material.

From the designers point of view it is the small strain amplitude

behaviour - involving severe softening - that is of great importance.

5.1.2 Effect of step changes on cyclic behaviour. Cycling can be looked

upon as being a particular method of cold working.a material. Thus

the strain history of a specimen might consist of a sequence of cycles

carried out at various amplitudes of plastic strain. Step tests, in

which specimens are cycled at a series of strain amplitudes, were

attempted by Wood and Davies (24). Some recent tests by Jo dean Morrow

Oiag, 7of (i)3 Illustrate the practical difficulties encountered in tests

of this sort. Sufficient cycles must be carried out at each strain

amplitude for the specimen to approaph:closely to the settled cyclic

state; but if too many cycles are applied the subsequent life of the

specimen is so shortened that few tests can be made at other strain

amplitudes. The choice of strain amplitudes for.use in the present

investigation proved to be a happy one. They were sufficiently large

to provide a rapid approach to the settled cyclic state but not so

large as to make the life•of the specimen unduly short. On the other

hand, they were not so small that the settled cyclic state could not

be achieved within a reasonable number of cycles.

The series of step tests conducted as part of the present research -117-

were all made on copper that was in the annealed state at the start of

the cyclic tests. No tests were made on material that had been init-

ially cold worked by uni-directional deformation because of the pro-

longed cycling that might have been needed for the first of the step

tests.

The results of the step tests together with those for the tests

made on copper that had been cold worked by various methods provide

strong evidence that the settled cyclic state for a particular strain

amplitude is independent of the previimis strain history, however complex

that may have been. That this is so - at least in the case of some

metals - is of great importance to designers. Designing structures

to withstand cyclic plastic deformation is never simple; but it is far

less complex if the,.calculations do not have to take into account the

complete strain history of the material.

Apart from the behaviour associated with the elimination of

the yield.step the softening of heat treated En25 steel appears to

take place in a similar manner to that of cold worked copper. The

tensile and compressive stress strain curves for hardened --:n25 steel

(Fig. 50 .) show that, unlike cold worked copper, the material is

symmetrical in its properties. Thus, in the case of steel any

asymmetry in the earlier stress-strain loops is produced by the

cycling process itself.

5.1.3 Cyclic behaviour of metallurgically unstable metal. The strain

cycling tests made on stainless steel showed that up to a certain -118-

number of cycles - the number depending on the strain amplitude used

- the fully soft stainless steel cyclically hardens in a manner similar to annealed copper. But, once the transition point is reached it starts to harden at a far greater rate. The new rate - like the old - is greater for,large strain amplitudes than for small. (The use of a logarithmic base for the number of cycles in Fig.56 tends to

Conceal this fact) The transition point occurs after many cycles when the strain amplitude is small and after few cycles when the strain amplitude is large. In the latter case the change of slope in the curves of Fig. 56 is slight and the point cannot be located accurately.

For the smaller strain amplitudes the material has evidently reached apseudG-settled cyclic state. Insofar as settled cyclic tests (see 4..8.1) were confined to the range of behaviour prior to the transition point the material appeared - like copper - to have a settled cyclic state dependent on the strain amplitude but independent of strain history.

It would appear that once the transition point has been passed cycling at successively lower ranges of strain amplitude will not soften the material. Presumably prior to the transition point being reached cyclic softening is possible.

If the oc•curence.of the transition point depends solely on the number of cycles as each strain amplitude then a designer can use the cyclic stress-strain curves of Fig.57 to design structures that are subjected to cycles of constant strain amplitude. The curves are of no use if therhaterial_is.no:t fully-soft,or:if there is a time effect. -119-

5.2 Survey of Cyclic Tests Between Fixed Limits of Stress

5.2.1 Effect of equal stress limits in tension and compression. The step

cycling tests, discussed in 5.1.2 above, indicate that if a material

is cycled between gradually changing limits of plastic strain the

stress limits will also gradually change in such a manner as to keep

the cyclic stress range at the appropriate value for the current strain

range. The strain cycling teas have also shown that the settled

cyclic stress 'limits always have 7ery nearly equal values for tension

and compression. Thus, if a material is cycled between equal fixed limits

of stress the strain amplitude.should tend to the limiting value

appropriate to that stress amplitude and the settled limits of cyclic

strain should remain very nearly constant.

Because of the large capacity for cyclic hardening possessed by

annealed copper it was difficult to verify this by tests made on a

single specimen cycled at a particular strain amplitude. The -results

of a test, however, showed an approximate agreement between the strain

and the stress cycling results. Aluminium proved slightly more satis-

factory to test and of the two specimens tested ore gave good agrement

and the other an approximate agreement between the strain cycling and

the stres3 cycling results. A stainless steel specimen tested at a

small amplitude of nominal stress - at which it could be expected

to achieve a settled cyclic state before a transition to rapid hardening

occurred — showed good agreement when the results of strain and stress cycling

tests were. compared. (Sen 4.6.2) i‘rether. specimen was tested ata cyuiie -120- • '

strain amplitude that was sufficiently high (+25 tons/in2) to ensure

that transition would commence before a settled cyclic state was reached.

This showed a rapid rise in hardness for the first few cycles. The

transition point was then reached and thereafter -faicwing a very slight fluctuation -a very gradual increase was observed. (See 4.8.2) At.first sight this seems strange when compared with the rapid hardening after the

transition point is passed in strain cycling. If, however, .a horizontal line (at 25 tons/in2 ) on the graph of Fig. ,156 is considered it will be realised the change (assuming the'no stress history effect rule holds)- consistsofatranslation from curve to' curve. On this basis many cycles are needed to produce a small reduction in strain amplitude.

In the cyclic tests between fixed limits of nominal stress it was found that with zero mean nominal stress a-gradual reduction_in the mean diameter of the specimen occured (cyclic creep). .Attempta to eliminate this by providing a small compressive mean stress were never completely successful. To stop the cyclic' creep, as Benham (31),

did, by making small adjustmentACto the stress -limits effective', but it is really an abandonment of stress cycling and a return to strain cycling: The cyclic creep is extremely sensitive to small variations in the mean stress and this makes investigation of cyclic creep at small values of mean stress very difficult. If one of the stress limits is varied - as was done by Benham - very precise control of the limit is essential. The nature of the load control system of the Instron testing machine used in the present tests made it possible -121-

to vary the mean nominal stress (by altering the position of the zero point on the chart) while leaving the machine's limit controls untouched.

Thus, small changes could be made in the mean stress while leaving the stress range exactly as it was before. Tests on annealed aluminium made using small mean nominal+3-ress values showed that a mean nominal compressive stress of about 0.06 tons/in' should cause zero cyclic creep. In a similar test on copper (using only ten§ile-meanstresses) a mean nominal compressive value 0.1 tons/in2 was found. It was pointed out by Benham that the change of cross section airing each cycle demands that

Since the inception of the present work an'infestigation ofLthis phenomena hawbedn'feported,bSmCoffin (1): His_results,forAtnnastiod aluminium; show the shift of the stress-strain loop at the 10th cycle plotted against the mean stress. His results show a sudden transition to occur from tensile creep to compressive creep for a small change of mean stress. The present investigatiowindioates-,a gnedualrchangeL fronuone dimantion:of-creep'to thea9ther.: Howbveri.the-quitmtity mastitis ured-was diffeteht :being: the aterage creep- pet cycle after a settled ' cyclic state had been achieved. A furthef point of‘differepce is.that step*tests.made on a single specimen were 'used in the.present work. -122-

The results of the present tests support the view that the

value of the nominal compressive stress needed to give zero creep

exceeds that required by geometrical considerations alone.

The fact that, however delicately the limits are adjusted, it:is

never possible to eliminate cyclic' creep entirely' is'not< solely due

to the exceedingly fine adjustment of the limits required. The condition. aimed at is, in fact, one of unstable equilibrium. Let it be imagined

the mean nominal stress is in error by an infinitesimally small amount.

A change' of;,.mestri specimen diameter will occur during each cycle causing an increase (or decrease) in the true stress range,and, hence,.in.the

strain range for each cycle. A change of strain range demands a change in the mean stress/strain range ratio and, with fixed limits of nominal

stress this does not occur.. Thus an infinitesimally small initial cyclic creep (of either sign) will be followed by creep at progressively increasing rate during the subsequent cyclea. The origin of this instability is the same as that tif the 'instability (described by coffin( )) *hich'haddes a ohange of profile f-adius during cycling.

It may well be that, if the limits are set with great accuracy, a very large number of cycles could be made before any appreciable creep was observed. Only at the stress ranges producing large amounts- of-cyclib deformation. will the cyclic.creep develop rapidly.

In the tests described in this section the strain amplitudes were small and the gradual build-up of the cyclic creep rate did not interfere with the observations during the small number of cycles applied. -123-

r..?.? Cycling between stress limits within the elastic range. Cyclic

hardening or softening must involve structural changes within the

material. These can hardly occur as a result of truly elastic defor-

mation; for this implies a complete return to the original form on

unloading and this cannot be reconciled with a change of internal

structure. Thus if a material behaved in a completely elastic manner

during stress cycling it would not be expected to harden or soften.

It is doubtful whether any material ever behaves in a perfectly elastic

manner under test - quite apart from anelastic behaviour - for any

minute local plastic deformation that occurs is masked by the elastic

behaviour of the main mass. of the material. Probably the softening

of steel under cycles of stress, as observed by Bairstow (17), is the

result of the extension of minute zones of plastic deformation;: likewise

the development of soft spots in cold drawn copper wire during fatigue

testing that_was observedi by Kenyon (47). With the limited number

of cycles used in the present tests softenipg of this kind is unlikely

to develop significantly. However it was felt that some tests should

be made using stress limits within the elastic range of the materials

to see if any sign of cyclic softening could be detected.

To provide the best opportunity for any softening to occur the

tests were made using stress ranges only a little less than the elastic

range. ilthough the results of tests made in both tension and comp-

ression on specimens of the materials in the states as tested Were-

available it was not possible to obtain very accurate values for -124-

the elastic limits from these. It was therefore not possible to obtain a very accurate value toritheslaStio'range, However, an estimate was made and this provided a satisfactory basis for deciding on suitable limits for the cyclic tests.

1A.th heat treated steel the elastic range proved to be symmetrically disposed about zero stress. With cold worked metals the elastic range is initially - unsymmetrical:-- with respect to zero stress. It had to be assumed that the first loading would change the elastic limits without altering the elastic range. The results of the tests justified this assumption.

Hardened and tempered En25 steel tested between equal limits of stress

(sue 4..7.2) behaved. in an entir4y elastic: manner for those stress ranges considerably less than the estimated elastic range but softened when cycled at a stress range slightly less' than the elastic range. The plastic deformation was observed to build up rapidly from a very small initial value. This is the behaviour that one would expect during the elimination of the yield step.

In the case of severely cold worked copper 100T:cycles at a stress range 'Jelievad to be a little: smaller.' than the. original elastic range. (see,

4.5.2 ,..p.85,) produced no softening but cycling at a stress range which was known to exceed' the original elastic range caused a softening of the material to occur. This was gradual at first but rapidly increased. Since it is the plastic deformation that causes the softening it is expected that when softening does occur it will -125-

(at first) develop rapidly. Cold drawn aluminium cycled for _6700 cycles at a stress range::thatwatapproximately the sameas the estimated elastic range. ( sett 24.. 6 . 2) showed a. 'Mall :cyalila. plastic deformation. at..the first cycle and and this did not increase during the subsequent cycles. The existence of a minute cyclic creep also indicated that the material was not in a completely elastic condition. It is not possible to differentiate between the very narrow loop produced by anelasticity - which may-dotl-be associated with softening - and the loop representing cyclic plastic deformation. The results for aluminium appear not to be in conflitt with-thobe-obtaihed-TrOttpoopper.

These tests reveal an important difference between the cyclic softening of metallurgically stable metals when tested between limits of plastic strain and limits of stress. In the former case cycling itt any selected strain range will reduce the stress range to the appropriate stable value; but in the latter case if the stress range is less than half the elastic range then cycling will have no softening effect. For it to occur the elastic range must first be decreased by cyclic softening carried out at a stress ampl&tude which exceeds half the original elastic range. In strain cycling softening occursfmost rapidlrat firstcand thenAhe-rate.decreases -as the settled cyclic state is approadhed lath'stress cycling .the softpning is slowat first and then the rateAncreases - because the greater plastic deformation enables softening to proceed more easily - but the rate must finally decrease to zero as the metal approaches the settled cyclic -126-

state appropriate Mo the particular range of stress. It is not to be expecteli that the material will soften to an unlimited extent.

Any slight initial non-uniformity of stress across the test section would give a false softening effect at stress ranges near to the elastic range. Bearing this in mind, it would appear that when metals that do not show a yield step are cycled between equal and opposite limits of stress giving a stress range less than the elastic range a small

(in terms of fatigue life) number of cycles will not produce softening of the material. If, however, the metal has a yield step, the stress cycling can soften it by the elimination of the yield step, in the manner described by Bairstow. At a high range of stress such softening can be quite rapid.

A similar investigation was made on cold worked copper (sec p85) using unequal limits of stress in tension and compression, the farmer twice the latter, and this supported the view that softening does not occur if the stress range does not exceed the elastic range. A further eyclic.test, made with stress limits of zero and a tensile maximum stress, just within the elastic range, (made on the same specimen) gave inconclusive.results-because - for reasons discussed later - cyclic creep began to occur in an uncontrolled manner.

Similar investigations (on another specimen of the same initial hardness) using a fixed maximum tensile limit, set just below the elastic limit, and a step by step decrease of the lower limit from a high value of tensile stress down to zero, showed no appreciable -127-

cyclic softening; even when there was complete removal of the stress.

In such partial or complete unloading tests th?development of a loop

in the unloading and reloading stress-strain curves was taken as an

indication of softening, while development in the rate of cyclic creep

was not.

The limited investigations carried out suggest that, for a material

that doesizat show a yield step, cycling between stress limits that

are within the elastic range for the first cycle does not cause

softening of the material. It must be emphasised, however, that the

numbers of cycles used in the present investigation are very small

compared to the expected fatigue life of the material.

5.2.3 Effect of large values of mean stress. The effect of small changes

in the mean stress during stress cycling upon the progressive defor-

mation (cyclic creep) of the specimen has been discussed in 5.2.1 above.

If the mean stress is increased the cyclic creep becomes greater at

first but presently decreases as the value of the mean stress approaches

the maximum stress - i.e. the case of ,stress cycling in which both

limits are of the same sign and the stress range small.

In cycling between unequal fixed limits of stress in which plastic

deformation ie4resent cyclic creep must occur. It is inconceivable

that the stress-strain diagram for each cycle should form a closed

loop and if the loop does not close then there must be a progressive

change in loop position from one cycle to the next.

The behaviour of materialAn this intermediate range was not -128-

dealt with during the present investigation. The large uni-directional

deformations involved cause great practical difficulties to be eneour-

'tered.

Some investigations were made on the behaviour of materials

under conditions in which the mean stress was high in relation to the

maximum stress and the plastic deformation small. In this case there

were grave practical difficulties of another kind. The rate of cyclic

creep prooved to be highly sensitive to minute variations in the

stress limits.

Preliminary tests demonstrated the unsatisfactory nature of

using nominal stress limits instead of true stress limits for testing.

The difficulties are analogous to those met in ordinary creep testing;

under nominal stress the creep behaviour in tension is unstable. • In spite of the above objection nominal stress limits - for

which the machine gave very precise control - were used in a number

of tests to investigate the creep of metals under conditions where

the the stress was cycled between a maximum tensile value and zero

or some intermediate value. To avoid very large initial straits

work-. harder specimens were used and, for the same reason it

was thought best to carry out the tests in tension.

The results of the tests suggested that, if it were not for the

geometrical instability, the cyclic creep rate would settle to a

stable value. "This stable value could be estimated from the results

of_ tests made, between,limits_of,.nominal stress. -129-

Cold drawn copper was tested between a maximum tensile stress limit (just above the elastic limit) and a sequence of minimum tensile r stress down to zero. (See 4.5.2 p.85.) In all cases a SMall cyclic creep was observed 1,ut no reversal of plastic defermatien. 1.1t it must cc, remembered that the displacement of the cyclic stress-strain diagram is more easile observed than the development of a cyclic loop.

The cyclic creep rate appeared rapidly to assume a stable value in sac} case.. Its value was small for 1/4 removal of load and only twice as great for 5/6 removal of load. For full removal of load it.rose to about ten times the -irst value.

Examination of the reversed loading stress-strain curves for copper

(Fig. je.) indicate that the unloading curve is straight and that no reversed plastic deformation occurs on unloading. Thus it may well be true that no cyclic loop develops during cycling under conditions of full or partial unloading. The cyclic creep must be caused by t:e rounding that always occurs at the top of a re-loading curve during any interrupted loading test. Sachs et al (48) investigated the rounding effect and it would appear that it is present even for very small reductions of load.

Small as it is, the precise nature of this phenomenon can have a very profound effect on the rate of cyclic creep.

Any material susceptible to ordinary time dependent creep would, in tests made with a non-zero mean true stress, be expected to have its cyclic creep rate also dependent on time. In the materials tested the time dependent creep rate is very small and, as might be expected. -130-

changes made in the strain rate during cyclic plastic deformation tests

caused no change in the shape of the stress-strain: curves for the

various materials. It might b! thought that the rounding of the re-

loading curve might be a time dependent phenomenon; but changes in

the strain rate made during cyclic testing between zero and a maximum

stress showed no apparent change in the rate of cyclic creep.

In the,only test made between:limits ,of zero and, maxiinum,tensile

true stress, on cold worked copper, although the cyclic creep rate

was, in general, constant there were occasional cycles in which a

relatively large strain occured. It is possible that these were due

to some, very- slight, intermittent softening of the material. However,

the possibility that they were caused by some minute delay in the

reversal of the machine cannot be ruled out.

5.3 The Bauchinger Effect and Cyclic Behaviour 5.3.1 The presence of the Bauchinger effect in a symmetrical cycle. .The

term Bauschinger effect is here used to mean the difference in shape

between a stress-strain curve determined in loading in one direction

to that obtained for loading in the opposite direction. For example,

referring to Fig. • 6.0, which shows a typical set of reversed loading

curves for a metal, the difference in shape between the curves OA

and BC will be termed the Bauchinger effect.

The Bauschinger effect is present in material that has been cold

worked by uni-directional deformation. It is also to be found in material

that has been cold worked by cyclic deformation. The fact that a -11i-

settled cyclic state, with the same shape of stress-strain curves for both tensile and compressive loading, has been reached does not

mean that the Bauchinger effect has been eliminated. To carry out

the two tests needed to determine the Bauchinger effect it is necessary

to stop cycling and reduce the load to zero. (Ideally two specimens

should be used; one for the tensile test, the other for the compressive

test.) It is convenient to interrupt the cycle at such a point that

unloading leaves the specimen at the point of zero strain. Fig. 52

illustrates the marked difference in the re-loading and reverse load-

ing curves observed in a test on a copper specimen. It is not essential

to stop cycling so as to leave the specimen with zero strain. .or

instance, the cycling can simply be stopped at one of the points of

zero stress. It is clear that the Bauchinger effect will again be

present in a pronounced form; the reloading curve consisting of a straight portion joined by a small radius to the continuation of

the old cyclic loading curve, while the reverse loading curve has a pronounced curvature from the start.

It might be thought that there is some point in the cycle at

which the specimen might he unloaded and then be found to have no

iauchinger effect. That this is not so can be demonstrated without recourse to experiment. Assume for simplicity - though the argument

does not require this - that the settled cyclic loop has a wholely

straight elastic line for unloading and a wholele curved elastic- plastic line for loading. Ilso thoueh this aL:51in is not vital to -132-

the argument - that, however the cycle is interrupted, the elastic range remains unaltered. To achieve a state of symmetry in the elastic behaviour of the material in tension and compression it is necessary to remove the load when the point in the cycle representing half the maximum stress is reached. Although the material now has equal elastic limits in tension and compression the Bauchinger effect (though not so severe as before) still exists. The tension and compression curves beyond the elastic range cannot be the same shape. This is illustrated diagramatically in Fig. 62, in which it will be observed :that one elastic-plastic curve is tangent to the elastic line while the other is not.

Theauschinger: effect Isalways_pretett in„a material...that has been,StreSS'or'strain_eyeled to a settlekeyeliet state;. but it .can vary in its form according to the manner in which the cycling was concluded. Thus the use of the stress-strain curve to obtain a_ measure of a materials hardness must be treated with some caution.

If the Bauschinger effect could be eliminated without altering the hardness of the material the resulting stress strain curves would provide a unique measure truly representative of the material's hardness. The possibility of eliminating the Bauchinger effect without changing the hardness of the material are discussed in 5.3.2 below.

If the stress-strain curve for each reversal.during cyclic deformation could be predicted then the complete set of stress-strain -133-

curves for any programme of cyclic deformation could be constructed and the strain hardening or softening effects and the cyclic creep predicted. It is therefore of interest to enquire whether the shape of the reversed load stress-strain curve follows any simple law. For simplicity the shape would have to depend cnly on the very recent strain history of the material. It cannot be assumed to depend solely on the value of the last applied, strain; for the origin for strain is an arbitrary one. It might more reasonably be assumed to depend (Solely) on the value of the last applied maximum stress. ( a 'recei, 4717 ) The last assumption was investiated by comparing the reversed loading stress strain curves for a series of specimens that had recieved a single reversal of stress with the corresponding curves for specimens that had recieved a number of stress reversals made at progressively increasing amplitudes of deformation. This investigation was made only in the case of copper, which only permitted comparison to be made up to a limited level of stress; for annealed copper can only be cold worked in simple tension to a moderate degree of hardness.

The equipment used in the present investigation was not entirely suitable for a detailed study of the Bauschinger effect; for small departures from the correct stress-strain curve which might occur , in the first cycle - and which had no appreciable effect on the results of cyclic tests - rendered some of the test results of little value for the present purpose. However, the curves of Fig. 28 certainly support the view that the shape of the reversed loading curve is -134- .

decided by the maximum value of the previously applied stress. Further

comparisons are made in Fig. 34 which shows results obtained from

(a) annealed copper,-(b) swaged copper and (c) hardened and tempered

En25 steel. The broken line curves are for material that had reached

a particular degree of hardness by strain cycling and the full line

curves are for annealed material that had reached corresponding degrees

of hardness by simple tension.

Although there is some support for the assumption that the

Bauschinger effect depends on the maximum,value..of_tbe laet applied;

maximum stress, yet this cannot beentirely without-qualification4

Clearly, if the last applied stress was so small that it caused only

elastic deformation it would not decide the subsequent behaviour of

the material. Also, if it were so small that the strain it caused

was small in relation to the strain caused by the previous stress

it could not be expected to entirely over-rule the effect of that

previous stress.

5.7-2 The removal of the Bauschinger effect by decreasing cycles. An

annealed specimen that has been subjected to cyclic plastic strain

of very small amplitude possesses only a slight Bauchinger effect.

Likewise a cyclically hardened specimen that has been reduced to the

same cyclic state by the application of a sequence_of cycles of

progressively decreasing amplitude possesses a slight Bauchinger

effect though, in its condition after the original cyclic hardening

it possessed a considerable Bauchiriger effect. One of the effects of -135-

the application of decreasing. cycles ofA)lastic deformation_ig, therefore, the elimination of the Bauschinger effect. This has been demonstrated in the course of the present work for both copper and En25 steel. Now, it has been shown that it cannot be eliminated by one

reversal of• aeformation and the theclose:lipPtoach:ofthematerial-tao

atateAll Which'Bausphinger effect. is absent - may_requitecmawcyclimi A very close approach to_symmetry.was obtained withEn25...steelAising, only-30 zycles.s_(See 4th-a:.material,that:_haSa.large Capacity for :cyclic hardening_ and sOftening-the_few-zYcleS:needed7to'xiliminate_the Bauschinger effect may well cause an appreciable softening of the material. This was shown to be the case with copper (see 4.5.3 p.51.)_, though with En25

it would appear that the Bausehinger effect had been removed without any change occuring in the general hardness of the material. En25 steel it appears that the cycling down process has left the elastic range unaltered, while in the case of copper the:

larger number of cycles caused an appreciable reduction in the elastic range. The degree of accuracy with which limits of elasticity can

be observed makes it difficult to bebertain'oh thii'lioint. But it suggests that - inconvenient as the measurement is in practice - it - may be that . the elastic range is a quantity that provides a measure ,of hardness independent of directional effects introduced by the Bauschinger effect.

It was not possible to make tests on the material in directions -136-

at right angles to the axis of the specimen but it is not to be supposed that the cycling down process, using stresses applied along a single axis, wilLhave• rendered the material isotropic. A specimen may have similar curves in tension and compression when tested along" one axis but different shapes of curve when tested along other axes.

Since the inception of the present work some similar tests, involving cycling up as well as down,'made.on annealed copper, and cold.worked copper have been reported by Morrow ( 1 ). lie-wishedto try to obtain the:,iettled cYclicittrest straincurvefrom- a:sequence of-tests.Joade on a. tingle specimen. If step tests can provide the data for plotting the curve - and, in the case of copper it has been shown'that they, can

- then it follows that a series of increasing cycles could be used. It is essential, however, that the increments be very small so that there are many cycles and the specimen is always near to its settled cyclic state. Morrow's autographic records (Fig. 14 of his paper.) indicate that he used a.large number of cycles for the first cycling up. Yet it seems they were not enough. Only when he repeated the cycling up and doWn processes twice was there some indication that a settled cyclic loop was being obtained for each strain amplitude; but this may not have been true, for his tests on cold worked copper did not show it to havez-reached- the same: cyclicstate for each strain amplitude that the annealed material had reached. In the present tests theL settle&cyclicloop:.-wai,:established fors he maxitumamplitude.atAhe

-coMmenCethent,:of:tha;tests'ane--it.wis also checked after each test. -157-

The use of progressively increasing cycles (termed by Morrow 'spectrum testing') to establish the cyclic stress strain curve should be quite satisfactory provided a sufficiently large number of cycles are used; but it appears that - particularly for small strain amplitudes

- this number should be very large indeed. The same results should be obtained by a cycling down process; though even larger numbers of cycles may be needed. The settled cyclic stress-strain curve is the curve that passes through the peaks of all the loops.

After his final cycling down Morrow carried out a simple tension test. He points out that the stress-strain curve for.this-liesvery near the settled cyclic stress-strain curve,Ahough slightly below it at high strains. The results for En25 steel (Fig.53 .) described in the pesent work show just the same relationship. But the results for annealed copper indicated that with prolonged cycling the simple tension curve is sitpated..well-below,the.peaks.. (Fig.,31 may be_looked upon as an illustration of this; though in this case'the cycling down was not gradual but was done in steps.) It would appear that the behaviour of the annealed copper is due to its ability to cyclically soften and' that:had MorroCcgrried out a larger number of cycles he would not have found such close agreement between the curves.

The fact that the stress-strain curve for annealed En25 steel after the cycling down process passes through the peaks of all the cyclic loops - each of which, in this material, can be taken as representing a stable cyclic state - suggests that this curve is -138-

simply part of the original tensile stress strain curve of the material

minus the yield step (which has been removed by cycling).

The fact that the corresponding curve for annealed copper lies

well abovethe original tensile curve (Fig.31 .) may lie in the fact

that'to completely lower the curve enormous numbers of cycles at small

strain amplitude are needed. 6oftening at small amplitudes is slow.

5.3.3 The prediction of cyclic behaviour from Bauschinger tests. If. the

Bauschinger effect for various stress levels has been established by

a series of reversed loading tests on different specimens a family

of curves of the kind shown diagra-atically in Fig.60 (a) may be drawn.

If the Bauschinger effect depends on the maximum value of the last

stress prior to stress reversal and not on the whole stress-strain

history then it is possible to predict the behaviour of the material

when cycled between fixed limits of plastic strain from the information

contained in these curves.

'Referring to Fig.60 (a), let it be supposed that the curve OAC*E*G,

is the stress-strain curve in simple tension for a material which

would cyclically strain harden. If a specimen were subjected to a

single reversal of plastic strain range BD it would traverse the path

OABCD, in which BC would be different to OA owing to the Bauchinger

effect and C would be higher than A. Next a specimen is loaded mono- * tonically in tension to C , which has the same stress value as C, and

the unloaded and reloaded in compression to E to give a plastic strain

range FD which is the same as BD. Again E will have a higher value -139-

than C owing to the Bauschinger behaviour. Let the process be repeated

on, say, two further specimens for the cyclic paths OE*F*GH and OG*H*IJ,

in each case keeping a constant plastic strain range during reversal, M * * i.e. JH = HF = BD and the stress levels for E and E, the same and

for G* and G the-Same. The curves• for Fig: 60 •(a) Eire- nOir.redrawn-io

giveFig.60.(b)._:Curve.0ABC'becomes O'A'g'C', curve C*D*EF is up-ended• * * to become C'D'E'F', curve E F GH becomes E'F'G'H' and-so on.

The Cyclic stress_ strain diagram-efFig.601;(b)-has a:`. similara

form to the cyclic stress-strain diagram for annealed material...(Fig.17.

It shows cyclic hardening to be occuring and the rate decreases from

cycle to cycle. In the Bauschinger curves illustrated in Fig.60 (a)

for the particular strain range examined above the Bauschinger effect

always gave a peak compressive stress greater than the peak tensile

stress that preceded it and this led to cyclic strain hardening. Now, s\-ctitc,/ if the peak compressive stress is greater than the peak tensile stress

that preceded it cyclic softening would result. If the difference

between the peak compressive and tensile stresses decrease as they

change in magnitude the material will harden or soften as explained

above, but the rate of hardening or softening will decrease with

cycling. If the difference between the peak compressive and tensile

stresses becomes zero, hardening or softening ceases.

In the above argument it is assumed, for the sake of simplicity,

that the tensile stress-strain curve is the same shape as the compressive stress-strain curve and the Bauschinger effect is the same for tension -140-

and compression. If this is not so the reasoning still holds.

Rather than use a series of specimens to produce a set of reverse loading curves it is better to use a single specimen and to subject it to a sequence of cycles of increasing amplitude. Fig. 29 shows the stress strain loops produced from a specimen of annealed copper. The use of a single specimen avoids the scatter in the results provided by a series of specimens; it permits higher stress levels to be reached than can be obtained in a simple tension test; and it does not necessitate the re-plotting of all the curves using true :.stress valpes. _The use of a Ainglespeoimen is permissible once it is established that the Bauschinger effect depends_solely on the maximum value of the last applied-.stress and not on previous history.

The actual construction of a set of cyclic stress-strain loops for a, particular strain amplitude, using the family of reversed loading curves is no easy matter. A more satisfactory check on the validity of the hypothesis is to attempt_to-construet:thp settled, cyclic stress- strain- curve from information'supp.lied by_the- family.of.reversed loading curves.

The settled eyelic stress-strain:eurve represents.the condition. for which cyclic hardening or softening becomes zero. Thus the points on it may be obtained from the reversed loading curves of Fig. 29 by a simple construction. Point:CAm - theIlcurviAlZ tarksAhe position where the stress on reverse loading reaches the same value- as the stress just before the reversal. Line CD is drawn parallel to the elastic T

-141-

unloading lines. Thus DB represents the amount of plastic strain

corresponding to the stress level represented by points A and C.

If strain cycling were carried out at this particular range of plastic

strain then hardening or softening would cease upon this particular,

level of stress being reached. Thus the construction supplies one

point on the settled cyclic stress-strain curve for copper.

Fig. 61 iS a reproduction of F:ig.23 showing the cyclic stress-

strain curves for annealed and cold worked copper with the points

obtained from Fig.29 added to it. At the smaller strain amplitudes

further points, obtained from reversed loading tests made on indiVidual

specimens, have been added. Except at small strain amplitudes these

points lie in the region between the two families of cyclic stress-

strain curves in about the position tht one would expect the settled

cyclic stress-strain curve to occupy.

The limitation imposed on the assumption that the Bauschinger

effect is independent of past strain history should not impose any

restriction when the cycling of annealed materials between fixed

limits of plastic strain is considered. ',,here it will apply is in

the first few.cycles when materials that have been initially cold worked

are cycled between fixed limits of plastic strain. It must be expected

that the effect of the original large plastic deformation will continue

to influence the Bauschinger effect for a number of cycles. Indeed,

the fact that the points in the curves for heavily cold worked copper

in Fir. 19 initially lie on separate branches of the curve shoW this to be so.

If the hypothesis is correct it is of considerable value to designers for it can be used to predict cyclic behaviour under any kind of.cyclic deformation - for example, cycling.with constantly changing limits of strain - subject of course to the above limitations.

A further test on the validity of the hypothesis is whether it will satisfactorily- predict the results of cyclic tests carried out between fixed limits of nominal stress. The theory predicts that, since the stress amplitude is fixed for each reversal the strain range cannot vary from cycle to cycle. In other words, the specimen should achive a settled cyclic state immediately aftthe-initial loading.

In the stress cycling tests made on Annealed oopper,and annealed aluminL ium this did not occur (see 4.5.2 and 4.6.2); the reason for this being that the first reversal of load produced a strain that was small in comp-

.arisson with the strain produced by the, initial application of load; which invalidated the assumption that the Bauschinger effect depended only on the maximum value of the last applied stress. That this would occur could be ascertained from the form of the reversed loading curves without actually carrying out cyclic tests.• However in a cyclic test on stainless steel made between fixed limits of nominal stress the specimen settled into a stable cyclic state after the first loading and showed little change (apart from some cyclic creep) thereafter. The pryperties of this material were such that:the reversed loading produced a strain that was of considerable magnitude compared -14.3-

to the strain caused by the initial loading. This, also, could have

been ascertained in advance by reference to the reversed loading curves

for the material. The stress cycling tests made on En25 steel Were

concerned with the softening due to the elimination of the yield step

so they have no bearing on the point under discussion.

_nalop:ue Models of Cyclic Behaviour r-.4.1 General. Analogue models of complex mechanical phenomena may be

or value in two ways: They may • - 4/9 in -the_case of_eleetrical-analogues

- provide a system on which experimental measurements are easily.

made; or they may be unsuitable for actual physical operation - as-are

the various mechanical models used in the study of creep - but provide

a better understanding of the phenomena under investigation and, in

particular, provide a fruitful source of suggestions as to_the.kind

behaviour that is to be expected in circumstances differing from

chose already investigated. It is from the latter point of view

that they are considered in the present work. Their essential requirement

is that they should be built up from simple units, which, though they

may be combined in a complex mariner, give a model whose behaviour can

readily be visualized.

Models consisting of pin-jointed bars. A material composed of

"rains of anisotropic material orientated at random is statically

indeterminate and it is therefor of interest to compare it to a

simple statically indeterminate structure. One of the simplest

that can be selected consists of three pin-jointed bars mounted as -1414-

shown in Fig. 63 ,a.ch bar is assumed to behave elastically up to a certain load and then to yield at constant load. A force, representing stress, is applied to the lower joint and the v(,rtical deflection of this joint represents strain. The model produces a simple form of stress-strain diagram as shown. This has an elastic portion followeed by rr :-astic-plastic portion in which strain hardening occurs at. a constant rate. it also illustrates the Bauschini7er effect and a cyclic stress-strain loop. A description of its behaviour ,;ilder various conditions is given by Timoshenko (:49). The addition of further bars will convert the elastic-plastic portion of the stress- strain diagram into a series of lines of successively decreasing; slope; thus, by a suitable choice of model properties, making the stress-strain diagram approximate to the diagram for.a real material.

A conversion of this model into the form of a spoked wheel (50) permits it to be used for. the study of tri-axial cyclic stress; but when the model is concerned with uni-axial stress -the use of inclined bars is not essential; bars with the requisite characteristics mounted in parallel serve as well.

Studies of the behaviour of single crystals have shown that the

Eauschinger effect is present in their stress-strain curves. Thus in a poly-crystalline material the bauchinger effect cannot be attributed solely to the introduction of ..a—syptem of intexnal.stresses when some grains yield before others during plastic deformation. Nevertheless the pin-jointed bar model is a useful form of analogue. •

5.4.3 Models consisting of springs and sliding blocks. A block,,aidingl

on a horizontal,surface againbt1Tourombfrj.ction and with a spring

attached to it, through which a force is applied„Trotrides.the

same characteristics as one of the bars described in the previous

section. A model composed of a set of such blocks mounted in parallel

waas constructed by:7Jenkin'in.1922'(51). Apart :from ther;fact that

this type of analogue can be constructed in a workable form,fit'has

a great advantage over the pin-jointed bar model in that the springs

provide a visual indication of the force in each element of thenmodel

at any instant of time. Thus, when the load is removed after a

sequence of deformations the new lengths of the springs will indicate

the nature of the internal stress distribution.

Vihen the form.mf,stress-strain diagram generated by a model

composed of two elements mounted in parallel„Fig.64j4),.is opmpax;ed

with that generated by a model composed of two elements in series,

Fig.64(b), it is found that, given suitable spring stiffness and

block friction values, the two diagrams are the same. Thus the

behaviour of the Jenkin - type-off model can be taken to be similar

to the behaviour of the more complex models madetwith the same %Maio

eletentS that have been proposed_ from time tá:time.'_

By the use of a sufficient number of spring and block elements

mounted in parallel it is possible to obtain a close approximation

to the shape of the tensile or compressiVestressrstrain.curvesfor

any given.material;provided that both the curves are of the: same. shape. Study of the behaviour of two block-models,- for which a selection.

of direct and reverse loading curves is shown in Fig. 65 , reveals that

the reversed loading curve is the same shape as the original stress-

strain curve for loading in that direction but that it is displaced

in position both vertically and sideways. The same must be true fpr a

multi-block model; because this can be built up by combining two block

models. It follows that a feature of the Bauschinger effect as port-

rayed by the spring-block model is that the elastic range (if any)

of the original material is unaffected by plastic deformation. If

the applied load exceeds a value of twice the the elastic limit of

the original material (assumed the same in tension and compression)

then plastic deformation will occur before unloading is ccmpleted.

This behaviour is quite unlike that found in the results of

the reverse loading tests on copper (Fig. 28 .) nor for.. any of.tbe other

metals tested. In the case of copper tY.e difference is particularly

striking. The annealed material has virtually no elastic range but

with stretching a large elastic range can be developed.

It would appear that two effects may be present in the strain hardening of metals and that in the ordinary tensile test the two effects are inseparable,. One is an intrinsic hardening - best represented by a rise in the elastic limit of the material; the other an extrinsic hardening due to the_Bauschinger effect - this increaslnif the apparent hardne.ss, (as. indicated by the_stress-strain curve) in the direction of deformation while decreasing it in the other direction. The spring-block-model apparently portrays the Bauschinger effect alone. The intrinsic hardening effect might be incorporated by arranging for the friction of the blocks to increase as the. deformation proceeded but this would destroy the simplicity of the model.

Despite these shortcomings is of interest to examine the cyclic behaviour of the model. Study of the two block model shows that both for cycling between fixed limits of strain and for cycling between fixed limits of stress the model_immediately(_gives a.stable-stress- strain loop. Some examples for stress and strain cycling on material in the original state and on material initially stretched are given in Fig. 65. In the case of strain cycling an initially cold worked material it appears that the model does not generate symmetrical limits 4 of stress. It ,could appear that the multi-block model shows a similar behaviour

The behaviour of the multi-block model is of particular interest

.en its conduct-under a cycling down process is examined for--it portrays faithfully the behaviour observed in the tests on En25 steel

(discussed in 5.3.2). the amplitude of cyclic strain is progressively decreased the amplitude of movement of each block decreases until it finally reaches its mean position and stops there. Thus the result of applying a large number of cycles of progressively decreasing amplitude to the model is that it is restored to its original state and contains no internal stresses. Its properties in tension and compression are then the same. -148-

4.4 Spring-block models with changing characteristics. Vihiteman in

1959 proposed a mathematical model for depicting the stress-strain

relationshjp (52); this model being equivalent to a multi-block model in

which all springs had the same stiffness and the blocks various friction

values. Sy an analysis of actual stress strain diagrams he derived

the requisite distribution of friction values needed to fit the model

to the experimental results. He suggested that the distribution varied

with past strain history. With such a model intrinsic hardening

could be introduced. The weakness of the model from the point of

view of the study of cyclic behaviour is that, using a fresh distribution

of friction values for each reversal, the model can be fitted to any

set of experimental stress-strain loops.

It'would be interesting to know how a model which had the requisite 4 distribution of of friction values to produce the original stress-strain

curve, but in which, at each reversal, the friction values were re-

allocated, at random, amongst the blocks would behave.

A mechanical model in which each block had different friction

values for each direction of motion could be constructed without great

difficulty but this would not show a progressive increase in hardness

from cycle to cycle.

To investigate the behaviour of the more complex models it would

be necessary to use a computer. If the mechanical model analogue were

replaced by an electrical analogue very complex systems could be studied

and the the unrealistic ones rejected. -149-

5.5 Cyclic Behaviour in the Light of Dislocation Theory. The Bauschinger effect in a poly-crystalline material composed

of perfect crystals is accounted for by the setLing up of internal

stress systems when the material is plastically deformed. The existence

of a Bauschinger effect in single crystal specimens is explained in

the same manner. All crystals contain inaperfectS440,of various kinds

.and. after plaztic'deformation has oceured.these imperfectioris

the cause of a system of residual stresses. 'Thus„,apart from .any change

in crystil*structure7_a_Bauschihger effect will be introduced.

The present investigation and other investigations on poly-

crystalline material as well as investigations on single crystal specimens

suggest that, in uni-directional deformation, there is an additional

hardening effect - here termed intrinsic hardening.

The behaviour of materials under cyclic deformation indicates

that in cyclic softening there is both extrinsic and intrinsic Battening.

The evidence for the latter is that cycling of a cold worked material

can reduce'the elastic range.

Although the extrinsic hardening is caused both by inter-crystalline

effects and by changes within the individual crystals the explanation

of intrinsic hardening must be sought within the crystals.

The strain hardening that occurs during plastic deformation of

Single crystals is now known to be by the interaction bf dislocations • with each other and with various kinds of barriers'Ahat exist. in the

crystal structure. In addition to the.moVementof dislocations -150-

the creation and destruction of dislocations can occur. Besides this, barriers, such as point 4efects can be created or destroyed or moved.

The picture, during uni-directional deformation, is constantly changihg but the pattern develops in a progressive manner. In cyclic deformation the changes are more complex. A dislocation moving one way will meet an entirely different set obstructions to those it meets when moving the other way. In one direction of motion a dislocation may pass out of a crystal, never to return. Dislocations of opposite sign that meet can eliminate each other, but this process cannot.be reversed.

Frank-Read source can act as a sink hut the process would not appear to be an exact reversal of the one by which it generates new dislocations.

In a mass of material these effects average out.and most materials have, after the first few cycles, very nearly the same behaviour during loading in each direction. Cyclic behaviour of a material is generally independent of the direction of the first applied load but it may well be that the first movement of an individual dislocation has a lasting effect on its subsequent movements.

•Thus the behaviour of dislocatiOns during hardening by uni- directional loading differs very much from their behaviour during cyclic hardening. It is for this reason that the rather complex analogue model described above (in might give a more satisfactory reproduction of the cyclic behaviour of real materials than is provided by the simpler models that have been described. -151-

6 CONCLUSIONS

Cyclic tension-compression tests made between fixed liMits of plastic

strain show that:

(1)OFHC copper will cyclically strain harden or soften according to

its initial state and the strain amplitude. The method by which the

material is cold worked has little influence on its cyclic hardening

or softening behaviour. Aluminium - on which a less extensive invest-

igation was conducted - appears to have a similar Cyclic behaviour..

Thus, for these materials there appears to be a settled cyclic state

for each plastic strain amplitude to which the Material tends during

cyclic deformation regardless of its initial mechanical properties.

It is not suggested that Phis rule applies to all metals 'that

have been work-hardened from the annealed state and then subjected

cyclic deformation. Tests on,00pper 7.5 aluminiurh alloy'reported- by

Feltner-and:,Laird-in 1966'(53) after_the,,completion of the present work

show that the settled cyclic state for this material can be very much

influenced by previous strain history. They suggest that only the

metals that show a wavy slip - amonf which are copper and aluminium

- possess settled cyclic states that do not depend on past strain

history.

(2)En25 steel always cyclically strain softens; though, except at

small amplitudes the softening of the annealed material is insignificant.

The degree of softening .increases with decrease in tempering temperature.

At the smaller. strain. amplitudes the softening that occurs is mostly due -152-

to the elimination of the yield step; a phenomenon first observed in

a 0.34;,C steel by Bairstow in 1910.

Austenitic stainless steel, which is a metallurgically unstable

material, when cycled at'small strain amplitudes at first displays

a similar behaviour to that observed in copper and aluminium; but

after the completion of a certain number of cycles, there is a transition

to a very rapid rate of hardening. The smaller the strain amplitude

the more the transition is delayed. With large amplitudes rapid

hardening appears to occur almost as soon as cycling commences.

Cyclic tension compression tests between fixed limits of nominal

stress show that:

(1) Annealed OFHC copper. and..aluminium subjected.to cyclic plastic

deformation using .equal limits of tensile and compressive- stress harden

towards 4.tettled.state-according to the'Stresaamplitude used. The

tests indicated that in the settled cyclic state the relationship

between stress amplitude and strain amplitude is the same regardless

of whether stress cycling or strain cycling is used. That this is so

supports the view that each settled cyclic state is independent of

the previous stress (or strain) history.

(2) Cold-worked OFHC copper and aluminium cycled between equal limits

of tensile and compressive stress show:no sign of softening and show

no appreciable departure from completely elaetic behaviour so long

as the stress range does not exceed the elastic,range in the initial

state.'however, the number of cycles appli.A .;(1 these tests was small -153-

compared to the expected fatigue life of the specimens. Similar tests made on En25 steel using a stress range within the elastic range show a softening of the material towards a settled cyclic state. This type of cyclic softening is the kind that was observed by Bairstow and is caused by the gradual elimination of the yield step phenomenon.

(3) The small cyclic creep found in tests made between equal limits of nominal tensile and compressive stress is shown to be not entirely due to the small tensile mean true stress caused by the cyclic, variation of the area;-.of: test' section. Both copper and aluminium appear to need a: very small zompressive mean true stress if cyclic , creep is not to occur. It is pointed out that the development of this cyclic creep is to be expected, however carefully the- limits are adjusted, because the state of zero creep is an unstable one.

(z) The cyclic creep that occurs when the nominal stress limits arer, both tensile shows an uncontrolled increase from cycle to cycle because of geometrical instability. A test conducted on cold worked copper using_true-stress limi4s—suggestIr that the _creep rate,x.apidly-:draws near to a.. settled value;, though- some fluctuations were observed to.occur from cycle to cycle. Tests on cold worked copper with only partial unloading in each cycle sugt;sst that cyclic creep occurs even when_ there is no trace of plastic deformation on removal of the load. The creep in each cycle appears to be associated with the rounding of the top of the re-loading line. It does not, however, appear to be sensitive to a change in the rate of cycling. -154-

Tension and compression tests madcaonmateriaLaftbrperiodsof cyclic testing shOW the.tiistence of'a-Bauschinger effect in the cycled rpfteriall. :It is shown that this .effect is present,.nomatter how the the cycling is terminated. Cycling down tests in which a material is subjected to a large number of cycles of plastic deformation of progressively decreasing amplitude „show that it is possible to eliminate the Bauschinger effect by mechanical means; but, in copper this cannot be achieved without a general softening of the material.

In cycling between fixed limits of plastic strain it is very conven- ient_to .use stress amplitude as a measure of the hardness of the material though'this quantity measures not only the general:hardness of the material but incorporates an additional apparent hardness which is due to thee,Bauschinger effect. This latter hardness - here, termed extrinsic hardness - can be reduced by a cycling down process. The tests on coppe1. indicate that, at the same time, the former type of hardness - here called intrinsic hardness = is reduCed. In annealed En25. tteelAhere:.isctvidence_that'dycling.'dowiLdbesnot:,16wertheintrinsic hardneds.. The.factthat the_elastic range forcopperis :greatly reduced by cycling down while that of steel appears to be unaltered suggests that the elastic range may looked upon as a measure of the intrinsic hardness.

An examination of mechanical analogues composed of springs and blocks which slide against Coulomb friction shows that, although these can be constructed to generate any shape of stress-strain curve in -155-

tension (or compression) the hardening effect they represent is the

extrinsic one only. They reproduce the Bauschinger effect and the

cyclic stress-strain loop but not the cyclic hardening or softening

effect. A more complicated model is suggested which,-it_is believed,

would introduce the intrinsic hardening effect,,whichj.s_responsible,

for the change_in shape of the stress-strain.loops as-pycling proceeds.

A method is proposed by means of, which.the cyclic strain hardening

and softening behaviour can be predicted using information obtained

from a series of reversed loading tests made to establish the Bauschinger

effect for various levels of stress. The success of the method depends

on whether the Bauschinger effect is influenced solely:by-the maximum

value of the last applied stress or whether it is dependent on the

entire previous strain history. .Subject tocertainlimitations-it-

0 does appear to be independent of previous strain history. Although the method appears capable of predicting the behaviour of a material

during strain cycling it is because of the aboVe limitatione; not

able to predict its behaviour in stress cycling.

Only a limited study of the,Bauschinger effect was possible in the

course of the present research. Thp.apparatus useddid_not permit a

detailed study of the shape of, the stress-strain Curve in the region. of

the elastic range. Nor withoutepparatus able to record:true stress,

•values - instead of nominal stress values - are tests at the large

deformations corresponding to high levels'of stress practicable. -156-

Table 1. Analysis of aluminium.

He i Cu Si .Mn .1 Zn Ti V i .33 0.01 I 0.09. 0.02 0.021 0.01 .4=0.01 <0.01 F 1

Table 2. .nalysis of steels.

C Mn Si S , P Ni Cr Mo A. 1 n25 0.32 0.58 0.19 0.016 .(-021 2.50 0.(,)5 0.52 - i Stainless 0.0 1.r,6 0.701 0.013 0.0314 ,E'',. '1..2 - -137-

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