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(Technicians, notably mechanical engineers, engineering draughtsmen and metallurgists, are trained in a technology of the mechanical properties of materials, often called "Strength of Materials", which is soon shown to be an anachronism in this modern age. It lacks either the unifying conception necessary in any system, a purposeful, rigid terminology, or any symbolism of its own: it is incoherent and even contradictory. Physicists and mathematicians in their occasional and independent approach to such matters We not subscribed to the weaknesses of the technician's outlook; but, on the other hand, they have never denounced the standard version, nor yet substituted order for chaos by building anew on a sound foundation.
This latter task has been the writer's private interest and personal concern during the last six years. With minor differences the following is to appear as an Appendix I in the Third Edition of his textbook "Essential Metallurgy for Engineers" (Pitman) which is now being prepared for publication. The footnotes refer to papers and articles by the writer since early in 1941.)*
INTRODUCTION**
"Strength of Materials"
In all probability the development of an ancient art into a modern technology rarely takes place smoothly, but
rather tends to proceed in strides-from the inception of aims to the perfection of their moans of attainment, then back to reconsideration of aims, and so on. It is tenable that development in engineering and metallurgy during the last fifty years has resulted in the carrying of "means of attainment" as far forward as they are likely to go with ease until a now stop is taken to reconsider "aims". For, by contrast with the energy and understanding put into actual operations by metal and other craftsmen of to-day, the fundamental standards by which they are still directed are characterized by no little weakness and misconception.
* The necessity for reform and the now basis of a suitable version were both urged by the writer in a letter to "The Engineer" of March 14th, 1941.
** Abstract from a paper in the Proceedings of the Institution of Mechanical Engineers (Received September 8th 1942).
At the root of the trouble with these basic standards lie lack of system, loose conceptions and vacillating terminology. From the systematic study of "Mechanics" we turn to "Strength of Materials" with something akin to dismay; for, instead of finding in it an allied type of applied mathematics, we may observe that all sight has been lost of the elementary necessity to retain Specific words as terms to convoy single rigid ideas; so that calculations are hardly possible, in fact the heaviest handicap is imposed at present upon the transmission, recording, and handling of the plain facts about materials.
In this technology, we find that the word "resistance" can mean unital force, or else a ratio (as in "creep resistance"), or else an amount of energy (as in "impact resistance"). "Strength" sometimes moans all the properties which are stresses; sometimes it even includes those measured as strains; sometimes it refers narrowly to a false, sometimes to a true, maximum stress. We never inquire too closely what "hardness" may be; but we put it aside by itself, and then speak of "strength and hardness", usually meaning by this, all the stress properties; we prefer to speak of "hardening" without referring in particular to any increase in the opposition to indentation, but merely to describe the raising of the stress properties in general. Each of us is permitted to work out independent ideas of "toughness" - or rather of "brittleness", because for some unknown reason the negative property has always been the more respectable of the two in this particular case. We have been told how complex and subtle "toughness" really is; and it certainly would be complex if it must necessarily be the average of whatever metallurgists and engineers have ever felt it to be! But why do we adopt this attitude? In Mechanics, "work" is an absolutely fixed and simple thing having definite dimensions with which no one may tamper; it is not permitted to be the average of all that so many interested persons have thought about its. "Ductility" is commonly defined in one way, and tested in another way, whereby we get a "nominal" or wrong result; there is often quite a definite feeling about this property that "softness" is involved - but not "weakness", because the latter work is permitted to convey feelings of uncertainty of behaviour rather than the simple opposite of strength.
To make further progress, it seems highly important that all concerned should for the time being cease to feel so many different things about the behaviour of the materials under loadings; at the moment we should adopt a precise terminology, and learn to handle it in an unequivocal way, if not in a mathematical way. Adequate visualization would then follow, and our thinking and feeling would then be in line with our perfectly practical tests.
THE SHAPE OF A MATERIAL'S REACTION TO FORCE
(Abstract from "Metallurgia", April, 1945.)
At one time a good many people thought the earth to be flat and, no doubt some became so steeped in the notion that difficulty was entailed in escaping from it - difficulty shared, of course, by any attempting to go to their assistance. The campaign of assistance to follow in those pages could be said to involve similar and perhaps even greater difficulties, comparable only with the above if, in pursuing the same analogy, a special supposition is made.
Let it be supposed that, in the past, people had not actually been "flat-earthists", seeing that they had never entertained - far less contemplated - the possibility of the earth having any shape at all! But, for all that, be it supposed that they had arrived at certain limited conclusions about the earth's weather, for example, beyond which they could make no further headway. In order to overcome such a deadlock, clarify the position and point the way to a fuller understanding, it might have become necessary to show that -
(a) the earth could and did possess shape;
(b) the existing ideas were tenable only on the assumption of a flat earth;
(c) more precise ideas and advancement of knowledge of the weather could only be reached by the more tenable assumption of a round earth.
Major difficulties would be presented if questions of shape were quite neglected in attempts to solve geognosic problems.
As a body, the technicians to whom the new approach to mechanical properties is meant to appeal are hardly aware yet that these properties can be considered as parts of a system, of a scheme of a particular shape; far less have they visualised the shape. On the whole, they have been viewing each property separately as little more than a "something" evaluated from an arbitrary test by calculation with measurements therein made. They resemble the above hypothetical people who tried to fathom weather problems without knowing that the earth had shape. Their superficial views serve immediate and useful ends; but these are limited ends which will load on to no progress and discovery.
It is, therefore, submitted that if engineers and metallurgists are as vitally concerned with the mechanical properties - the reactions of materials to force - as in fact they are known to be, it must be vital for the future progress of their technologies that they should appreciate the system of relationships existing between the properties, the fundamental shape of the scheme of which the properties are but closely related individuals. If it be reflected that such technologists might be as heavily handicapped nowadays as electricians would be without their exact system of units related by Ohm's and other laws, then it would seem to be worth while to investigate the possibilities of a much more satisfying basis than that upon which "Strength of Materials" now stands.
THE PROPOSED SYSTEM
The proposed system, like so many others of potential and heuristic value, is simple. Technicians have been plotting "nominal" stress-strain curves, and perusing so-called stress-strain diagrams for a long time but have yet to see how one suitable stress-strain curve is a complete statement of all mechanical properties of the elementary kind, whilst certain other properties - hardness, fatigue endurance, notch-toughness and damping capacity - are derivatives or compounds of the same. In the electrical analogy - current, voltage resistance, etc., are elemental and are compounded in heating effect, electromagnetic effect, etc.
Four or five obstacles of widely different kinds have hitherto blocked the road to a better understanding, their very diversity accounting for the fact that one or other, or a combination of these difficulties has proved too great it. the past for the formulation of any simplification of the whole subject. To describe these difficulties is to explain a good deal.
CONCEPTIONS OF PROPERTIES
Probably the most widely experienced difficulty has been the general impression that the conceptions of the properties, such as of elasticity, hardness and toughness, have in themselves a cardinal or elemental value; and that, to be adequate, the tests should provide results in accordance with this implicit belief in the primary significance of certain
preconceived ideas and mental pictures. Such a viewpoint has been unfortunate. A quality like hardness may be appreciated and identified through personal reactions sufficiently well for the mental picture, and for the planning of the practical test to elicit (what is described as) "hardness". But, what can thus be known of hardness as the reaction by materials to that particular type of loading? Obviously nothing. Only upon examination of the mechanism of the reaction can hardness as a property of materials be understood at all. On examination, the mechanism of any test may prove surprisingly simple, resembling some familiar mechanism; on the other hand it may prove to be complex and incomprehensible except through its mathematics. In fact, the original conceptions of the properties, having served their purposes in suggesting types of test, have been outlived. The analyses of these tests have frequently pointed out the inadequacy - sometimes even the inherent falsity - of the very conceptions which gave birth to the tests. The conceptions have been the probes by the use of which considerable
advances have been made in understanding the reactions of materials to different kinds of force application; the understanding, or rather the reactions themselves, are the only mechanical properties of any significance for the future; the "probes" may now be forgotten.
Yet, oddly enough, the above difficulty of approach to the matter has not applied in every case. Some of the properties have no traditional background, and minds have remained unbiased, happily fancy-free, towards them. The proof stresses, for example, and fatigue endurances are properties around which minds have weaved no spell; they and some others have always been just what the mechanism of their tests prove them to be. It seems to have been agreed that a material's fatigue endurance cannot boar such resemblance to a man's endurance as to be dismissed as something quite familiar. Examination of this mechanism has revealed much; and "fatigue endurance" conveys nothing to the mind but the revealed mechanism.
TENSILE, COMPRESSION AND SHEAR TESTS
The second obstacle is by no means subtle; it involves no psychological difficulty of approach, and seems to call for a few well-chosen words which shall not mince matters. It might be described as the result of an ingenuous, if not a culpably artless, approach to considerations of stress and strain. The tensile, compressive and shear tests to destruction are carried out by recording the deformation taking place as the load steadily increases, by calculating the stress from the load and area, and the strain from the deformation and length, and by plotting the stress against the strain to obtain stress-strain curves. If stress and strain are deliberately miscalculated for the sake of convenience, or for any of the reasons given in extenuation of such a crime as the "nominal" type of curve, then the curves derived from tensile, compressive and shear tests have very different shapes.
If stresses and strains are calculated properly, however, then the curves for tension, compression and shear are quite similar in shape, and conviction grows that experimental errors and/or purely secondary reasons (of comparative unimportance to the generalisation itself) must be responsible for the small differences between them.
LOADING DIFFERENCES
The third obstacle is a matter of detail; it is the difficulty that there are these small differences between the curves of the three typos of loading, even if stresses and strains are calculated properly.
It has been appreciated that the results of the compressive test are incorrect if lubrication is not provided between the ends of the specimen and the jaws of the machine, friction at those points being obviously undesirable and likely to yield erratic results. Sheet lead is sometimes introduced at those points for lubrication. But this is only one recognition of a general truth which has not been so generally recognised - that, in all these tests which are to be comparable, the lateral strains must be entirely unrestricted. It can be seen that the shapes of those stress-strain curves will be profoundly affected by restriction of lateral strain. In fact; the normal long, low curve of the ductile material may even become the steep, straight line compressibility curve if sufficient restriction of lateral strain is applied. Therefore, no compressive test curve is exactly comparable with a tensile test curve unless its specimen is literally unrestricted in its lateral deformation. Moreover, in the torsional shear test, radial fibre shear is never unrestricted, and only becomes relatively unrestricted in tests on thin tubular sections.
Attention to details, of which the above are only two out of several possible causes of differences in shape, would be found to remove this particular obstacle. Homogeneity and isotropy have to be assumed in any generalisation and these are necessarily lacking in some well-known materials.
MEASURING STRAIN
Obstacle No. 4 is as serious as any. The one true stress-strain curve obtained by the removal of the last two obstacles approaches the shape of the true characteristic curve of the elementary mechanical properties, but suffers from one embarrassing feature. The area underneath the characteristic curve should, undoubtedly, represent the work of reaction by the material per unit of its volume.
Unfortunately, the area under the curve obtained up to the present does not; it can be shown experimentally that it does not, and examination of the actual units employed shows that it cannot. The difference between the correct work of reaction per unit volume and that estimated from the usual *1 curve may be as much as 50 percent or so.
The difficulty here is that the common method of regarding and of measuring strain happens to be unsuitable; it can be overcome quite simply by changing the method, and the more common conception of strain, to one which is not so common and perhaps a little tiresome at first. As usually conceived, strain is (L Lo)/Lo, or deformation by unit of original length; but, as this can only apply properly up to maximum uniform deformation, the complication in the case of a specimen with necks or bulges is adequately looked after by the formula (Ao - A)/A applied to the neck or bulge. The conception of strain is identical in each formula, and it will not satisfy the work reaction requirement of the true characteristic curve.
The type of strain required is that which may be described as "differential" or "logarithmic." It is the sum total of a vast number of infinitely small changes in deformation each divided by the length actually undergoing deformation. Where ordinary strain is s, or (A0 - A)/A, this differential strain is 2.3 log (1 + s), or 2.3 log (Ao/A).*2 This less common kind of strain is simply related to the common kind, and there can be no more object to making such a change than in changing from a fractional basis to a percentage basis, or from density to specific gravity, or from centigrade to Absolute degrees of temperature. If desired, the usual kind of strain can still be used in measurements of ductility, or for any purpose except that of measuring the specific work reaction of a material. The correct work reaction can be expressed by quite a simple formula in terms of the usual strain; no disturbance is strictly necessary, though it would be advisable to become accustomed to the idea of this more satisfactory type of strain.
*1 This refers to the most enlightened curve hitherto employing (A0-A)/A strain and in reality far from "usual". This true toughness may be four or five times that shown under the "nominal" (or most usual) curve.
*2 Abstract from "Engineering" June 2nd, 1944.
TOUGHNESS, AND HARDNESS
The fifth, and perhaps the final obstacle to the acceptance of one true stress-strain curve to stand for the mechanical properties, is the undeniable fact that all the mechanical properties are not exhibited on such a curve. What, then, are toughness, hardness, notch-toughness, and fatigue endurance?
It may have been seen that toughness has just been dealt with; it is, in fact, the specific work of reaction by work per unit volume, up to any desired point of stress and strain, up to fracture point if desired. It is similar to proof resilience, the latter being mainly elastic toughness, while plastic toughness carries on beyond the "elastic limit," and a progressively smaller part of it is reversible energy. Toughness is the first compound property, and it can be road direct from the curve employing differential strain.
COMPOUND PROPERTIES
Meanwhile, there are numbers of tests for materials, specimens and special machine parts which have not yielded figures of merit ranking high as standard properties. Attention is drawn to one of the most elementary of these, because light can be thrown on the whole question of the compound properties by examining the following simple case -
A coiled spring is an example of a special machine part which may be required to undergo a test for, say compressibility. Now, the mechanics of the spring's action is known; lot it be supposed that it is, in fact, quite accurately known, and that a perfect stress-strain curve of that material is at hand. In such a case the test itself should be quite unnecessary, because the compressibility of that spring could be calculated with ease.
In the same way, there can be no compound property of which the mechanics, or mechanism, is really understood that cannot be calculated with the aid of the true curve of the material. From such a rule there would be no need to exclude fatigue endurance, hardness and notch-toughness if the mechanisms of the material's reactions to these types of loading were really known. When the above mentioned properties can be calculated, they should give greater satisfaction than they do now; for, by the time they can be calculated, they will undoubtedly be understood.
Toughness,*
for example, may now be calculated from the stresses (f) and the logarithmic strains (a) by measuring the area under the true curve thus -
∑ s = s f • ds s = o
and, if the curve shape can be identified by same such formula as f = s • sm, toughness can be calculated without area measurement from -
This seems to be an exemplary case. If toughness means specific reaction in work per unit volume, and if it is agreed that the area under the true curve measured that property, there can be nothing further to be said about toughness. For, it is a property clearly seen to be compounded from the elements of the true curve in a certain manner. By the criterion of the ability to synthesise a compound property from its elements, toughness is seen to be completely understood.*
* Abstract from "Engineering", June 2nd, 1944.
Toughness and Notch Toughness
(Abstract from Proceedings of the Institution of Mechanical Engineers, September 1944 (Paper receiver September 8th 1942).)
Simple considerations of thermo-dynamics suggest that the amount of energy necessary to overcome the cohesion in, or otherwise to sever unit volume of, a metal must be the sane whatever form of energy is employed, whatever the number and variety of stages adopted, and whatever the time taken in the task - as long as energy is neither wasted to surroundings nor strays into the system from surroundings. Since the energy to effect severance of unit volume by heat is the same as that required to melt unit volume, this amount of energy must be the specific heat multiplied by the temperature-rise to the melting point plus the latent heat of fusion. Expressed in work units, this energy should represent the ideal true ultimate toughness.
The true ultimate toughness of a metal when obtained from a mechanical test, must depend however upon the efficiency of the utilization of energy in the test. The laws of thermodynamics invoked for the above purpose provide for loss neither by conduction and radiation, nor by any physical change other than the destruction of the lattice, nor by any chemical change. For the sake of brevity, all these interferences must be referred to jointly in the category of "loss". The more rapid the application of work in the tests, the loss will be this loss since it takes place chiefly by conduction and radiation in the standard case, and the more will the true toughness correspond to the amount of energy as calculated from the heat properties - specific heat, molting point, and latent heat of fusion. It will be smaller, then, at the higher rate of loading, approaching the minimum or heat-calculated quantity at the highest rates; it will be larger at the lower rates, since a more protracted test will result in greater loss.
Toughness has been treated hitherto as though synthesized from the more fundamental stress and strain which are both known to vary with loading rate. Considerations of thermodynamics suggest the identity of ideal true ultimate toughness with the total heat needed to melt unit volume, showing that the energy to fracture is as fundamental and primary as it could be - even though it may be capable of analysis into stress and strain. If ideal true toughness is viewed as the fundamental constant minimum for any particular pure metal or alloy, it will be possible to see why variations in loading rate do in fact produce the known effects on stress and strain.
The inertia of the atoms (small masses under large cohesive forces) will account for the higher stresses of the higher loading rates; whereupon the principles of thermodynamics step in and call for lower strains in order that total energy should remain constant. When, at very high loading rates, inertia has maximum stressing effect but loss is very low, the high stress-low strain product will be very low, approaching the heat-calculated "adiabatic" type of change. When, at very low loading rates, inertia has minimum stressing effect but loss is maximum, the low stress-high strain product will be very high because of the "isothermal" nature of the change in which much energy is lost to surroundings.
True mechanical toughness must, then, depend on the loading rate: in any test, the true product of stress and strain at any stage must vary With the time of the test. Expressed in a different way: as loading rate falls from, infinity, strain varies (inversely and) faster than stress, owing to the smaller efficiency of the lower loading rates in the process of severing the metal. The true toughnesses of the metals are therefore worth having; for they can only be the same as the toughness calculated from the heat properties at an infinitely high loading rate, being greater than this on account of the energy loss in the mechanical test. Space does not permit of reference here to the elastic increment of toughness, or to any of the other attractive developments of this theme.
Notch toughness is now seen to be quite a different matter. This is no unital energy to fracture the metal with an excessively high loading rate developed at a notch; for, whilst the total energy to fracture is taken as the measure of the notch toughness, any examination of the localities of fracture behind the notch will reveal that very different volumes of the metal are involved in the fracture (see Fig.2). So large are these differences that by far the most important consideration in these tests must be the volumes which become involves in these fractures specific energy falling into the background in the great majority of cases.
Hence, a notch-tough metal is merely one causing a spreading of the energy beyond the notch - as effected by grain refining. A notch-sensitive metal is one in which the spread of the energy is inhibited either on account of coarse grain, or owing to casual or inherent internal notches, or else because the metal contains some potential mechanical, physical, or chemical energy by which it rips itself as soon as the excessively high loading rate at the notch acts as a trigger to release that energy.
As between two metals with equal capacity for spreading the energy of impact, evidently the tougher of the two will prove to be the mere notch-tough. Ordinary toughness will therefore play some part occasionally; being the deciding factor in any case which is doubtful from the standpoint of strict notch toughness. At the higher rate of loading - and in particular at the exceptionally high rate obtained at the notch itself - this ordinary toughness should be lower than that of the tensile test, and even more difficult to calculate than that of the tensile test.
Hardness *
It is now becoming possible to do rather more than conjecture how hardness may be synthesised if it is viewed in a forthright manner as strength, elastic and/or plastic axial stress reaction, greater than that of the true curve on account of the lateral strain restriction characteristic of different materials subjected to local or central loading. A recent analysis of the Brinell test shows that, under hardness conditions of lateral strain restriction, the ordinary strength (the stress of the true curve) is increased by the factor (
m - l)/(m - 2) where m is Poisson's Ratio, or even when it is an apparent Poisson's Ratio in plasticity.*
*(Abstract from "The Philosophical Magazine", November, 1944.)
In the case of hardness loading, for any given degree of axial strain induced in the material, indentation strain for example, the corresponding axial stress is (m - l)/(m - 2) times as high as that to be expected from the true stress-strain curve. The published values of m apply to what is known as the "elastic range" only, as Poisson's Ratio is supposed to be an elastic property. Any considerable plasticity causes this ratio of the axial train to the lateral strain to fall to a limiting value of 2 in the fully plastic or liquid state. As m falls, the value of (m - l)/(m - 2) rises, and presents a complication into which there is no necessity to go here and now. For, the true curve is also what has been known as the "strain hardening" curve; and, by the time any stress of the plastic range has been induced for a second or two, this strain hardening has so altered the stress-strain characteristics of the material that the comparatively straight part of the curve - the so-called "elastic range" - has risen to this erstwhile plastic stress. The published values of m are applicable once again, since the material's "elastic limit" has risen to the value of the plastic stress sufficiently for their use.
Hence, if a maximum strain (in single loading), a fracture or rupture strain, is induced in compression at the surface of a stool by the indentation of a very hard ball, the "strain hardening" - or (more generally speaking) the strain strengthening - has raised the "elastic limit" of the material to the fracture stress, say, 65 tons sq.in., at least sufficiently for the use (without appreciable error) of the usual value of m, which is 3.3 for the steels. In this case the factor (m - l)/(m - 2) becomes 1.77. Therefore, the axial stress required under hardness loading conditions to produce this effect of fracture is 65 x 1.77 or 115 tons sq. in., or 180 kg.sq.mm., which is the Brinell Hardness Number itself for such a steel, the hardness stress just at the surface where the maximum possible strain (by single axial loading) has been induced.
Similarly, if at some depth below the Brinell ball some strain, such as that of a 25 tons sq.in. "elastic limit" is found to be induced, then the relevant axial stress- to which the Brinell number has degenerated at the depth in question must be 25 x 1.77, or 44 tons sq.in., or 69 kg.sq.mm.
Thus, hardness may be synthesised from the elements of the true curve, and there would appear to be no further necessity to apply tests for it. Alternatively, it could be said, perhaps, that the only point in carrying out a hardness test to-day is that the test, a very simple one in itself, provides a compounding of m and of (single axial loading) fracture stress, neither of which are too well-known to-day, because they do not happen to have been carefully determined. Were the true curve and good values of m both available, there could be no more point in carrying out the hardness test than there is in carrying out a toughness test on a material for which a true curve is available.
Toughness and hardness have both been demonstrated, then, as examples of compound properties capable of being calculated like the compressibility of a coiled spring. It can be surmised with fair accuracy that fatigue endurance, damping capacity, notch toughness and any of the lessor lights amongst the properties will similarly lend themselves to calculation from the elements of the true curve.
Turning, now, to remaining properties which cannot be read direct from the curve, they appear to be compounded in one way or another from the elements of the true curve. Neither fatigue endurance, nor notch-toughness are true "unital" properties, as their figures of merit do not relate to lengths, areas, or volumes in the same unequivocal sense in which all the elementary properties and toughness relate.
The measure of fatigue endurance is certainly the fatigue "limit" which is a stress. Yet, this stress of the S-N curve is of a hypothetical nature, not directly applicable to the exact localities of failure, but rather to a superficial area of maximum stress at the critical section where failure localities are to be found. At the precise localities of failure, the elemental properties apply honestly enough; but it is only the statistical average effect of the operation of these elements at a number of separated localities that is reflected in this hypothetical "limit". Moreover, failure follows a great number of repetitions of such stress induction, and for this reason alone the latter is as different from the stresses in the standard tests of (single) tension, compression and shear as anything could be.
The measurement of fatigue endurance as a "limiting stress" is a device which serves a very useful purpose; but, like the "ultimate tensile stress", has no reality as a unital property of materials, and should be regarded as nominal. It is not improbable that fatigue endurance will in time be measured in terms of energy and volume as a property compounded from the elements of the curve in some way at present uncertain.
Notch-toughness is measured in foot-pounds of work par specimen tested; yet no attempt is made to estimate the part-volume of the test piece actually becoming involved in the failure. Useful as the Izod and other impact tests are for purposes of comparison, for fooling the way in the dark, no one is to know how much more useful it would be to understand the property sufficiently, to have it on a unital basis.
ONE TRUE CURVE
The one true curve to which reference has so frequently been made is the graph obtained on testing at room temperatures and at a rate of loading which involves the fracture of the specimen in a minute or so. Now, the loads, deformations, lengths and areas whose measurements permit the construction of the true stress-strain curve have not usually been measured with a high degree of accuracy, because the materials tested have lacked the high degree of homogeneity and "repeatability" which would make it worth while to measure accurately. In consequence, comparatively large changes of temperature or rate of applying load in these tests will pass unnoticed in the shape of the stress-strain curve of a material. Yet, literally, the shape of the true curve can only be of fixed shape at exactly one temperature and exactly one rate of applying the load. The shape depends, then, on both temperature and loading rate, or time; and this fact would be perfectly apparent in the testing carried out under slight variations of temperature and time, but for the fact that the measurements made are not careful enough, nor is the available material "repeatable" enough. Whenever such testing is carried out at temperatures and times greatly differing from the normal, differences in the curve shapes become apparent in spite of crude materials and measurements.
In general, the effects of temperature and time on the curve shape is as follows - high temperatures and "high" times (slow loading rate) make the curve flatter and longer, and vice versa. There are some complications in the important cases of iron, copper and their alloys. It would seem to be possible to ascertain the shapes of a material's curves at a variety of temperatures and rates of loading, and to express differences in those shapes by means of some kind of coefficient to be applied to a standard curve shape. This might not be as difficult in practice as it sounds; because it may be found possible to apply a logarithmic formula to the curve shape, and the variation in shape with temperature and time change could then be indicated by changing values of k and of n in such a logarithmic formula as f = k • sn.
Even somewhat rough approximations to the mechanical properties at different temperatures and times would be much better than none at all. Up to the present very little attention has been paid to determinations of the curve shape, the graph-base of all the mechanical properties, at temperatures and times greatly differing from those which have come to be considered "normal". Some curves showing how individual properties change with changing temperature are available; and a little work has been done in the direction of discovering the effect of a changing loading rate. The new outlook on the properties suggests that the reasonable thing to do is to provide a standard stress-strain curve with coefficients by which to calculate from it the shape under conditions (abnormal in testing) which may be quite commonplace in forging, rolling, pressing, extruding, drawing, and so on.
ELASTICITY
The study of curves of certain crystals has suggested the literal reality of an elastic range terminating in a limit of proportionality and/or an elastic limit. However this may be, there is no material of construction like this; in spite of the fact that wrought iron and mild steel superficially appear to be elastic up to points in their curves where a discontinuity occurs and plasticity begins, it is true to say that all materials react to force with a combination of elastic and plastic reaction from the beginning of the curve to the end.
For a few important practical purposes, of course, there may be nothing but usefulness in assuming the truth of Hooke's Law for those up to some limit of proportionality, and in assuming some elastic limit for a material; in such cases there may be a practical assumption without any literal reality in these limits. But the literal reality asserts itself in no uncertain way when liberties are taken with it, when attempts are made to discover the stage of strain amounting to "mechanical failure". It is then high time to point out that, in reality for materials, there is neither limit of proportionality, nor constant modulus, nor elastic limit, nor point where plasticity commences.
Nor, for that matter can "mechanical failure" be said to take place after a specific amount of work has been done on the material, at same proof resiliences. For, "proof resilience" must also disappear with the "elastic limit".
Probably the best course for any disposed to think that there is a very strongly marked "elastic limit" in the case of some materials (such as the steels) is to examine a stress-strain curve of one of the steels, for example, with an extremely open strain-scale, as can be obtained by the use of an electron extensometer. Then, what looks, on the more usual strain-scale as the "straight line" to the "elastic limit" is seen to be a curve, upon which it is well-nigh impossible to place a "limit of proportionality" or an "elastic limit". The more sensitive and accurate the device for measuring the so-called "elastic" deformation of any material, the greater the conviction that there is no literally pure elasticity, and no stress, strain or work at which mechanical "failure" could be said to begin. Nor is there a greater degree of hope in any suggestion that "failure" could be considered to be at the sudden yield of the steels.
SHORT SUMMARY
Present Style Mechanical Properties
(Unrelated, unsystematic and faulty in conception)
New (Intermediate Style Mechanical Properties
(All closely related by the true stress-strain curve)
Elementary Properties - Stress properties
Proof Stresses (at any desired strains, or at all (strains)
Proof Moduli (tangents of the chord-angles at any strains)
True Fracture Stress (the Proof Stress at fracture strain)
Elementary Properties - Strain properties
Poisson's Ratio, m.
True Ductility (logo Ao/ A strain for final or intermediate true stresses).
Creap Resistance (True Ductility at very low loading rate, and (often) at some elevated temperature, starting at some selected stress induction).
Compound Properties
Toughness (area under the true curve using logo Ao/ A strain for final or intermediate true stresses)
Proof Resilience (same as toughness, but in the predominantly elastic straining up to some ,selected proof stress)
Hardness (for any given degree of axial strain induced by loading of the hardness type, the corresponding axial stress is (m-1)/(m-2) times as high as that to be expected from the true stress-strain curve).
Notch Toughness as a "property" has been explained. Fatigue Resistance and Damping Capacity (analyses incomplete).
Final Style Mechanical Properties
(related by three constants for each material)
Required - True Stress-Strain Curve, f = k•sn and also Poisson's Ratio m.
The mechanical properties of a material are then k, n and three constants only which themselves change only for temperature and rate of loading. Those constants, determined for a sufficient range of temperatures and loading rates can then be made to serve for every purpose now required and also for a great range of purposes hitherto unexplored.
Examples of Use 0.1% Proof Stress ..............k• (0.001)n
0.1% Proof Modulus...........k• (0.001)n-1
Toughness (to fracture) k• snn+1/(n+1)
where s is the strain at fracture, or the true ductility at fracture; for s can be substituted ‑
(f/k)1/n
Proof Resilience (at some near-elastic stress such as k•(0.001)n
= k•(0.001)n+1/(n+1)
Hardness = f × (m-1)/(m-2), or k•sn × (m-1)/(m-2)
(This is the sane as the Brinell Hardness Number, except that the B.H.N. is always quoted in Kilograms/mm², if the f or s in the expression refer to the material's fracture point. In the correct conception, "hardness" is nothing more than the stress reaction of a material loaded to any degree but centrally only, i.e. not universally over the whole cross-section of the specimen, with a sufficient excess of unloaded material around the area of load application to effect the material's characteristic maximum restriction of the lateral strain of the load. The above expression means that, in all such "hardness"-type loading, the stress induced for any degree of strain developed is (m-1)/(m-2) times that to be expected from the true stress strain curve, i.e. from the curve in which no restriction of lateral strain is tolerated.)
The Applicability of the Expression f = k•sn
If the specimen's size is sufficiently great in comparison with the size of its inhomogeneities and/or its areas of anisotropy, and if the material suffers little or no sporadic or irregular physical change in consequence of loading, its whole stress-strain curve can be expressed quite closely by f = k.sn where k and n are constants for any given temperature and loading rate. A typical example of sporadic physical change on loading is to be found at the commencement of the curves of certain irons and steels in common use. This is a change which ceases, or is strongly chocked, at the point commonly known as to "jogged yield point". Such curves start with a f = k'•sn' formula and continue with a f = k"•sn" formula.
If a substance falls short in homogeneity or in isotropy or suffers irregular physical change during loading within the range of stress induction necessary for some practical application, then it seems fair to exclude it from the category of "materials" for that application. This provision does not exclude either cast iron or stool in common use because neither is applied for the exercise of its mechanical properties within the range of its eccentricities in behaviour.
It should be noted that the expression f = k•sn would fit oven the most erratic of the true stress-strain curves sufficiently well to enable much more correct determinations of the plastic properties of those materials to be made than has hitherto been possible. This is because none of the plastic properties have been determined by the present-style mechanical properties with any approach to accuracy.
Fatigue Endurance
The "fatigue limit", which is the present measure of
fatigue endurance, is the maximum stress at the critical section which will permit of about ten million reversals without failure. In the case of steels, this maximum stress permitting ten million reversals will also permit of same hundreds of millions without failure: it can therefore be called the "safe stress" for the steels. In the case of most metals, however, such a safe stress does not exist; the "fatigue limit" drops with increasing reversals as far as these S-N curves have been traced.
An enormous amount of work has been done on this subject; and it has become clear that in the mechanism of repeated light loading there is propagation of the minute discontinuities in the material, at the critical section of maximum stress induction, whereby they join up to form cracks which in turn join to produce the fatigue fissure finally responsible for the specimen's collapse under gross overload.
The chief point here is that no one doubts but that the material, thus seeming to fail at so low a stress as a fatigue "limit", in reality does nothing of the kind, It fails only at its true fracture stress at the edges of its discontinuities, cracks and fissure where enormous leverage magnification factors are known to exist which can be calculated. The writer has reached an intermediate stage in his own study of the propagation of these discontinuities, which suggests that oven the lowest stresses would seem to have their propagating effect (in sufficient numbers of repetitions) on even the strongest materials.
Undoubtedly, therefore, fatigue endurance is another compound property, a derivative of the true stress-strain curve. As soon as its analysis is complete it should be possible to synthesize it from the elements of the true stress-strain curve like the others without going to the trouble of making the test.
Damping Capacity
Under vibration, or repetitions of load reversal at high
speed in the near-elastic range of stress, materials absorb very different amounts of energy for repetition; those absorbing more are said to have higher damping capacity than those absorbing less for they tend to damp oscillations which would otherwise tend to build up to induce really damaging stresses.
The hysteresis loop upon which this phenomenon depends is completely dependent on the shape of the true stress-strain curve. This latest of the properties to be recognised as important is very closely related to the true stress-strain curve; and a 'Proof Resilience" would almost certainly be an adequate measure of it at the relevant loading rate.
Symbolism
In his paper received by the Institution of Mechanical Engineers in 1943 the writer made some tentative suggestions for a symbolism to be employed with his proposed New (Intermediate) Style Mechanical Properties. These met with a
good measure of approval. A simple system would be necessary
before calculations could conveniently be employed; and a committee would no doubt decide upon the symbolism as soon as necessary.
Present Significance and Future Potentialities
The writer's personal advantage in having lived with the new idea for six years has enabled him to assess the likelihood of immediate usefulness and the hopes for the future of those proposals.
Engineers and metallurgists have founded an enormous edifice upon the old foundations: building still continues unchecked. Yet, we know well enough that great structures were built in past centuries with still poorer tools than those we possess today. We look back in wonder at the efficacy of the ancient methods, even as future generations will marvel that our present structures could have been built with "Strength of Materials" in such a state of chaos.
Much as the substitution of the decimal system in our weights and measures and money would be desirable if it could conveniently be effected, we know that it will only creep in as occasion permits. But, the very fact that the decimal system is now available, for use as occasion demands, is of potential usefulness; it is a system which has come to stay by for any application demanded of it.
There is something more, however, in the new ideas on the mechanical properties than there is in the decimal system. The former has unearthed real criteria of ductility and of toughness which had not hitherto been appreciated at all. It has also shown how hardness can be calculated instead of measured, thereby indicating the true nature of hardness. In performing that service it has demonstrated the nature of the compound properties and suggested the necessity for a proper analysis of fatigue endurance to complete the picture. It has thrown light on elasticity, on the so-called point of mechanical failure, and on many other old beliefs.
The new picture which has been drawn of the effect of changing temperature and loading rate on the stress-strain curve clearly and simply indicates the nature of the changes in properties normally to be expected under these different conditions. The vital generalisation proved to exist with regard to these mechanical properties must ultimately throw a flood of now light in all directions.
Its Effect on Metallurgy
Metallurgical treatment is designed to effect changes in the mechanical properties, i.e. in the height and in the length of the true stress-strain curve. The writer's publications in the technical press have already shown this,*
and has classified all known metallurgical treatments by their influence on change in the curve shape.
*Realism in Metallurgical Treatment" (in Metallurgia). "A Logical Approach to Metallurgical Treatmenet" (in Metallurgia).
