by J E Gordon
One consequence of these facts is, I think, fairly well known to most archers, at least in a practical sort of way. This is that one must never, never, never ‘shoot’ a bow or a catapult without a proper arrow or other appropriate missile. If this is attempted, then there is no safe way of getting rid of the stored strain energy, and, not only may the bow be broken, but the archer will very possibly be hurt as well.
Resilience or bounciness
A wet sheet and a flowing sea,
A wind that follows fast
And fills the white and rustling sail
And bends the gallant mast.
Allan Cunningham, A Wet Sheet and a Flowing Sea
When Galileo settled down at Arcetri in 1633 to work on elasticity, one of the first questions he asked himself was ‘ What are the factors which affect the strength of a rope or a rod when it is pulled? Does the strength depend, for instance, upon the length of the rope?’ Elementary experiments showed that the force or weight which is needed to break a uniform rope by pulling on it steadily is unaffected by how long it is. This result is what we should expect from common sense, but the news has been some time in getting around and one still meets quite a number of people who are convinced that a long piece of string is ‘stronger’ than a short one.
Of course these people are not just being silly, for it all depends on what you mean by ‘stronger’. The steady force or pull required to break a long string will indeed be the same as that needed to break a short one, but the long string will stretch further before it breaks and it will therefore require more energy to break it, even though the force which is applied and the stress which is in the material remain the same. Put in a slightly different way, a long string will cushion a sudden blow by stretching elastically under the load, so that the transient forces and stresses which result are reduced. In other words it acts rather like the suspension of a car.
Thus in a situation where the load is jerky a long string may well be effectively ‘stronger’ than a short one. This is why the bodies of eighteenth-century carriages were frequently slung from the chassis by means of very long leather straps, which were better able than short ones to resist the jolts imposed by eighteenth-century roads. Again, anchor cables and tow ropes generally break, not from a steady load, but from sudden jerks, and so it is generally better to arrange for them to be as long as possible. Those who are liable to encounter large dry-docks or oil-rigs under tow at sea at night or in thick weather do well to bear in mind that each of the tugs is probably towing by means of nearly a mile of steel wire. These nautical processions therefore cover an enormous area of sea and can be terrifying to the casual seafarer.*
This quality of being able to store strain energy and deflect elastically under a load without breaking is called ‘resilience’, and it is a very valuable characteristic in a structure. Resilience may be defined as ‘the amount of strain energy which can be stored in a structure without causing permanent damage to it’.
Of course, in order to get resilience, it is not necessary to use a very long rope, such as a wire cable. It is often convenient to use much shorter members, such as the helical springs which are used in the buffers of railway trains, or pads of soft material such as are used for ships’ fenders, or materials of low Young’s modulus, like the foamed rubbers or plastics which are often used for packaging delicate apparatus. Such things are frequently able to stretch or contract much more in relation to their length and so store more strain energy per unit volume. The excellence of the suspensions of skiers and animals is due, in part, to the comparatively low moduli and high extensions of tendon and other tissues.
On the other hand, although low stiffness and high extensibility promote energy absorption, and so make it more difficult to break a structure by means of a blow, it is only too easy to make a structure which is too floppy for its purpose. This usually limits the amount of resilience which can be designed into a structure. Things like aeroplanes and buildings and tools and weapons have to be pretty rigid in order to do their job. In this respect most structures have to be a compromise between stiffness and strength and resilience, and the achievement of the best compromise is likely to tax the skill of a designer severely.
The optimum condition may vary, not only between different types and classes of structures, but also between different parts of the same structure. In this respect Nature is at an advantage since she has at her disposal an enormous range of elastic properties in the different biological tissues. A simple but interesting example occurs in an ordinary spider’s web. The web is subject to impact loads arising from flies blundering into it, and the energy of these blows must be absorbed by the resilience of the threads. It turns out that the long radial threads, which form the main load-carrying part of the structure, are three times as stiff as the shorter circumferential threads which have the duty of actually catching the flies.
Naturally, there are many other ways of storing strain energy and getting resilience than by using tension members, such as ropes or spider’s threads, or compression members, such as railway buffers and ships’ fenders. Any shape of structure which is capable of being deflected elastically will have much the same effect. Probably the commonest arrangement is to absorb energy by bending, like bows and gallant masts. This is what happens in plants and trees and in most car springs. High-quality swords are expected to be able to recover, elastically, after they have been bent so that the tip touches the hilt.
Strain energy as the cause of tensile fracture
Starting aside like a broken bow.
Psalm 78
A reasonable amount of resilience is an essential quality in any structure; otherwise it would be unable to absorb the energy of a blow. Up to a point, the more resilient a structure is the better. Such highly sophisticated devices as the Viking ships and the American horse buggy were very flexible and resilient indeed. As long as they are not grossly overloaded, such structures will recover when the load is taken off and all will be well. However, if we do overload them, then of course sooner or later they will break.
Now to break any material in tension a crack must spread right across it. However, to create a new crack requires a supply of energy – as we shall shortly see – and this energy has to come from somewhere. As we have said, it is quite possible to break a bow by ‘shooting’ it without an arrow. What happens is that the strain energy which was stored in the bow can no longer be disposed of safely as kinetic energy in the arrow, and so some of it is employed in producing cracks within the material of the bow itself. In other words the bow has used its own strain energy to destroy itself. The broken bow is, however, only a special case of all kinds of fracture.
All elastic substances which are under load contain greater or less amounts of strain energy, and this strain energy is always potentially available for the self-destructive process which we call ‘ fracture’. In other words the stored-up strain energy or resilience may be used to pay the energy-price of propagating a crack through the structure and so causing it to break. In a resilient structure there may be a lot of strain energy around, and the same sort of energy which the Romans used to batter down the massive walls of Carthage can equally well be employed to enable a super-tanker to break herself into two halves.
According to the modern view of the subject, when we break a structure by loading it in tension, we ought not to regard fracture as being caused directly by the action of the applied load pulling on the chemical bonds between the atoms in the material. That is to say, it is not the consequence of the simple action of a tensile stress as the classical text-books would have us believe.* The direct result of increasing the load on the structure is only to cause more strain energy to be stored within its material. The sixty-four thousand dollar question whether the structure actually breaks at any particular juncture depends upon whether or not it is possible for this strain energy to be converted into fracture energy so as to create a new crack.
Modern fracture mechanics is therefore less conc
erned with forces and stresses than with how, why, where and when strain energy can be turned into fracture energy. Of course, in simple cases like ropes and rods the classical concept of a critical breaking stress is usually an adequate guide, but in large or complicated structures, such as bridges or ships or pressure vessels, it has proved to be a dangerous oversimplification, as we have seen. What comes out of recent theory is that, whether a structure is subjected to a sudden blow or to a steady load, tensile fracture depends chiefly upon:
The price in terms of energy which has to be paid in order to create a new crack.
The amount of strain energy which is likely to become available to pay this price.
The size and shape of the worst hole or crack or defect in the structure.
The fact that the amount of energy required to break any given cross-section of material varies very greatly indeed between different solids is easily confirmed, for instance by hitting first a glass jar and then a tin-can with a hammer. The quantity of energy required to break a given cross-section of a material defines its ‘toughness*, which is nowadays more often called its ‘fracture energy’ or ‘work of fracture*. This property is quite different to and separate from the ‘tensile strength’ of the material, which is defined as the stress (not the energy) needed to break the solid. The toughness or work of fracture of a material has a very important effect upon the practical strength of a structure – especially a large one. For this reason we must spend a little time in talking about the work of fracture of various kinds of solids.
Fracture energy or ‘work of fracture’
Since, when a solid is broken in tension, at least one crack must be made to spread right across the material, so as to divide it into two parts, at least two new surfaces will have to be created which did not exist before fracture. In order to tear the material apart in this way and produce these new surfaces it is necessary to have broken all the chemical bonds which previously held the two surfaces together.
The quantity of energy which is needed to break most kinds of chemical bonds is well known – at least to chemists – and it turns out that, for most of the structural solids with which we are concerned in technology, the total energy needed to break all the bonds on any one plane or cross-section* is very much the same and does not differ widely from 1 Joule per square metre.
When we are dealing with the range of materials which are, rather understandably, called ‘brittle solids’ – which includes stone and brick and glass and pottery – this is nearly all the energy we have to provide in order to cause fracture. As a matter of fact, 1 J/m2 is really rather a pathetically small amount of energy. It is a sobering thought that, on the simplest theory, the strain energy which could be stored in one kilogram of tendon would ‘pay’ for the production of 2,500 square metres (over half an acre) of broken glass surface – which accounts for the effects of bulls in china-shops. This is why a bricklayer can break a brick neatly in half with a light tap from his trowel and it is why we have only to be a little clumsy in order to break a plate or a tumbler.
Naturally, this is the reason why, if we can possibly avoid it, we do not use ‘brittle solids’ in applications where they are in tension. These materials are brittle not, primarily, because they have low tensile strengths – that is to say they need a low force to break them – but rather because it needs only a low energy to break them.
The technical and biological materials which are actually used in tension, and used with comparative safety, all require a great deal more energy in order to produce a new fracture surface. In other words, the ‘work of fracture’ is very much higher – enormously higher – than is the case with brittle solids. For a practical tough material the work of fracture usually lies between 103 J/m2 and 106 J/m2. Thus the energy which is needed to cause fracture in wrought iron or mild steel may be about a million times as high as that needed to break the equivalent cross-section of glass or pottery, although the static tensile strengths of these materials are not very different. This is why a table of’tensile strengths’, such as Table 2, (p. 56), can be a highly misleading document when it comes to the choice of a material for a particular service. It is also why the classical theory of elasticity, based mainly on forces and stresses, which has been laboriously evolved over hundreds of years – and still more laboriously taught to students – is really inadequate, by itself, to predict the behaviour of real materials and structures.
TABLE 4
Very approximate figures for the work of fracture and tensile strengths of some common solids
Material Approximate work of fracture J/m2 Approximate tensile strength (nominal) MN/m2
Glass, pottery 1-10 170
Cement, brick, stone 3-40 4
Polyester and epoxy resins 100 50
Nylon, polythene 1,000 150-600
Bones, teeth 1,000 200
Wood 10,000 100
Mild steel 100,000-1,000,000 400
High tensile steel 10,000 1,000
Although the detailed mechanisms whereby such enormous amounts of energy can be absorbed within tough materials as ‘work of fracture’ are often subtle and complicated, the broad principle is really very simple. In a ‘brittle’ solid the work done during fracture is virtually confined to that which is needed to break the chemical bonds at, or very near to, the new fracture surface. As we have seen, this energy is small and amounts only to about 1 J/m2. In a tough material, although the strength and the energy of any individual bond remains the same, the fine structure of the material is disturbed to a very much greater depth during the breaking process. In fact it may be disturbed to a depth of well over a centimetre: that is, to a depth of about 50 million atoms below the visible fracture surface. Thus if only one in fifty of these atomic bonds is broken during the process of disturbance then the work of fracture – the energy needed to produce the new surface – will be increased a millionfold, which, as we have seen, is about what really does happen. In this way molecules living deep within the interior of the material are able to absorb energy and to play their part in resisting fracture.
Figure 9. A typical stress-strain curve for a ductile metal such as mild steel. The shaded area is related to the work of fracture of the metal.
The high work of fracture of the soft metals is primarily due to the fact that these materials are ‘ductile’. This means that, when they are pulled in tension, the stress-strain curve departs from Hooke’s law at quite a moderate stress, after which the metal deforms plastically, rather like plasticine (Figure 9). When a rod or sheet of such a metal is broken in tension the material is pulled out before it breaks after the fashion of treacle or chewing gum; the broken ends will then be tapered or conical and will look rather like Figure 10. This form of fracture is often called ‘necking’.
Figure 10. The work of fracture is proportional to the volume of metal plastically distorted, i.e. to the shaded area, and thus is roughly as t2. Hence the work of fracture of thin sheet may be very low.
Necking and similar forms of ductile fracture can take place because a great many of the innumerable layers of atoms in the metal crystals are enabled to slide over each other by means of what is called the ‘dislocation mechanism’. Dislocations not only enable layers of atoms to slide over each other like a pack of cards but they also absorb energy – quite a lot of energy. The result of all this slipping and sliding and stretching in the crystals is that the metal is enabled to distort and a great deal of energy is got rid of.
The dislocation mechanism,* which was originally postulated by Sir Geoffrey Taylor in 1934, has been the subject of intensive academic research over the last thirty years. It turns out to be an extraordinarily subtle and complicated affair. What takes place inside so apparently simple a thing as a piece of metal seems to be quite as clever as many of the mechanisms in living biological tissues. Yet the funny thing is that this clever mechanism cannot possibly be purposive, if only because Nature has nothing, so to speak, to gain from it, since she never makes any structural
use of metals, which very seldom occur native in the metallic state anyway. However this may be, dislocations in metals have been of enormous benefit to engineers and might almost have been invented for their benefit, since they not only result in metals being tough but also enable them to be forged and worked and hardened.
Artificial plastics and fibrous composites have other work of fracture mechanisms which are quite different from those in metals but which are fairly effective. Biological materials seem to have developed methods of achieving high works of fracture which are very cunning indeed. That in timber, for instance, is exceptionally efficient, and the work of fracture of wood is, weight for weight, better than that of most steels.†
Let us now go on to discuss how the strain energy in a resilient structure manages to get turned into work of fracture. If you like, what is the real reason why things break?
Griffith – or how to live with cracks and stress concentrations
Ony rollin’s better than pitchin’ wi’ superfeecial cracks in the tail-shaft.
Rudyard Kipling, Bread upon the Waters (1895)
As we said at the beginning of this chapter, all technological structures contain cracks and scratches and holes and other defects; ships and bridges and aircraft wings are liable to all sorts of accidental dents and abrasions and we have to learn to live with them as safely as may be, in spite of the fact that, according to Inglis, the local stress at many of these defects may be well above the official breaking stress of the material.