by J E Gordon
How and why we are generally able to live with these high stresses without catastrophe was propounded by A. A. Griffith (1893-1963) in a paper which he published in 1920, just twenty-five years after Kipling’s splendid story about a crack. Since Griffith was only a young man in 1920, practically nobody paid any attention. In any case Griffith’s approach to the whole problem of fracture by way of energy, rather than force and stress, was not only new at the time but was quite foreign to the climate of engineering thinking, then and for many years afterwards. Even nowadays too many engineers do not really understand what Griffith’s theory is all about.
What Griffith was saying was this. Looked at from the energy point of view, Inglis’s stress concentration is simply a mechanism (like a zip-fastener) for converting strain energy into fracture energy, just as an electric motor is simply a mechanism for converting electrical energy into mechanical work or a tin-opener is simply a mechanism for using muscular energy to cut through a tin. None of these mechanisms will work unless it is continually supplied with energy of the right sort. The stress concentration is quite good at its job but, if it is to keep on prising the atoms of a material apart, then it needs to be kept fed with strain energy. If the supply of strain energy dries up, the fracture process stops.
Now consider a piece of elastic material which is stretched and then clamped at both ends so that, for the present, no mechanical energy can get in or out. Thus we have a closed system containing just so much strain energy.
If a crack is to propagate through this stretched material then the necessary work of fracture will have to be paid for in energy and the terms are strictly cash. If, for convenience, we consider our specimen to be a plate of material one unit thick, then the energy bill will be WL where W = work of fracture and L = length of crack. Note that this is an energy debt, an item on the debit side of the energy account, although as a matter of fact no credit is given. This debit increases linearly or as the first power of the length, L, of the crack.
This energy has to be found immediately from internal resources, and since we are dealing with a closed system, it can only come from some relaxation of strain energy within the system. In other words, somewhere in the specimen the stress must be diminished.
This can occur because the crack will gape a little under stress and thus the material immediately behind the crack surfaces is relaxed (Figure 11). Roughly speaking, two triangular areas – which are shaded in the diagram – will give up strain energy. As one might expect, whatever the length, L, of the crack, these triangles will keep roughly the same proportions, and so their areas will increase as the square of the crack length, i.e. as L2. Thus the strain energy release will increase as L2.
Figure 11. (a) Unstrained material.
(b) Material strained and rigidly clamped. No energy can get in or out of the system.
(c) Clamped material is now cracked. Dotted areas relax and give up strain energy, which is now available to propagate the crack still further.
Thus the core of the whole Griffith principle is that, while the energy debt of the crack increases only as L, its energy credit increases as L2. The consequences of this are shown graphically in Figure 12. O A represents the increased energy requirement as the crack extends, and it is a straight line. OB represents the energy released as the crack propagates, and it is a parabola. The net energy balance is the sum of these two effects and is represented by OC.
Figure 12. Griffith energy release, or why things go pop.
Up to the point X the whole system is consuming energy; beyond point X energy begins to be released. It follows that there is a critical crack length, which we might refer to as Lgy which is called the ‘critical Griffith crack length’. Cracks shorter than this are safe and stable and will not normally extend; cracks longer than Lg are self-propagating and very dangerous.* Such cracks spread faster and faster through the material and inevitably lead to an ‘explosive’, noisy and alarming failure. The structure will end with a bang, not a whimper, and very possibly with a funeral.
The most important consequence of all this is that, even if the local stress at the crack tip is very high – even if it is much higher than the ‘official* tensile strength ofthe material – the structure is still safe and will not break so long as no crack or other opening is longer than the critical length Lg. It is this principle which gives us our main defence against undue alarm and despondency about Inglis’s stress concentrations. This is why holes and cracks and scratches are not even more dangerous than they are.
Naturally we want to be able to calculate Lg numerically. As it turns out, for straightforward conditions, this is much simpler than we have any reasonable right to expect. Although the mathe-^ matical process by which Griffith got there might be regarded as slightly alarming, the actual finished result is disarmingly simple, indeed brilliantly simple, for
or, put algebraically,
where W = work of fracture in J/m2 for each surface
E = Young’s modulus in Newtons/m2
s = average tensile stress in material near the crack (taking no account of stress concentrations) in New-tons/m2
Lg = critical crack length in metres.
(Caution, Newtons, not Meganewtons.)
So the length of a safe crack depends simply upon the ratio of the value of the work of fracture to that of the strain energy stored in the material – in other words it might be considered as inversely proportional to the ‘resilience’ In general, the higher the resilience the shorter the crack one can afford to put up with. This is another example of not being able to have things both ways.
As we have seen, rubber will store a great deal of strain energy. However, its work of fracture is quite low and so the critical crack length, Lg, for stretched rubber is quite short, usually a fraction of a millimetre. This is why, when we stick a pin into a blown-up balloon, it bursts with a very satisfactory pop. Thus, although rubber is highly resilient and will stretch a long way before it breaks, when it does break, it breaks in a brittle manner, very much like glass.
One solution to the problem of how to be resilient and also tough is that provided by cloth and basket-work and wooden ships and horse-drawn vehicles. In these things the joints are more or less loose and flexible and so energy is absorbed in friction -that is what all the squeaks are about. However, although hedges and birds’ nests are pretty resistant to attack, this way of going about things is not often used by modern engineers, except perhaps in the case of car tyres, where the rubber is saved from being unduly brittle by the incorporation of canvas and cords.
It will be seen that Lg shortens very rapidly as the stress, s, increases. Thus, if we want to be able to accommodate a good long crack safely at a reasonably high stress, we need the highest possible value of W> the work of fracture, in a good stiff material, i.e. one with a high E. It is just because mild steel combines good work of fracture with high stiffness, and is also fairly cheap, that it is so widely used and so important economically and politically.
Although, as we shall see, there are a lot of snags about applying the Griffith equation, which we have just described, and we ought not to regard it as a sort of God-given answer to all design problems, it does in fact do a lot to clarify various structural situations which used to be very obscure and full of mumbo-jumbo.
For instance, instead of messing about with thoroughly bogus ‘factors of safety’, one can nowadays simply try to design a structure to accommodate a crack of pre-determined length without breaking. The crack length chosen has to be related to the size of the structure and also to the probable service and inspection conditions. Where human life is concerned it is clearly desirable that a ‘safe’ crack should be long enough to be visible to a bored and rather stupid inspector working in a bad light on a Friday afternoon.
In a really large structure, such as a ship or a bridge, we probably want to be able to put up with a crack at least 1-2 metres long with safety. If we suppose that we want to plan for a crack 1 metre long, the
n, making the rather conservative assumption that the work of fracture of the steel is 105 J/m2, we find that such a crack will be stable up to a stress of 110 MN/m2 or 15,000 p.s.i. However, if we want to play safer and plan for a crack 2 metres long, then we shall have to reduce the stress to about 80 MN/m2 or 11,000 p.s.i.
In fact 11,000 p.s.i. is just about the sort of stress to which large structures are often designed, and in mild steel this stress affords a factor of safety (strictly speaking, what is called a ‘stress factor’) of between five and six – for what that is worth. As an example of the sort of way this works out in practice, of 4,694 ships subjected to routine inspection in dock, 1,289, or just over a quarter, were found to have serious cracks in the main hull structure – after which, of course, remedial action was taken. The number which actually broke in two at sea, though still too high, was something like one in five hundred, a fairly small proportion. If these ships had been designed to a higher stress, or made of more brittle material, in most cases the cracks would not have been spotted before the ships broke at sea and were lost.
According to the pure and simple Griffith doctrine, a crack shorter than the critical length should not be able to extend at all, and therefore, since all cracks must start life by being short ones, nothing should ever break. In fact, of course, for all sorts of good reasons which are the affair of metallurgists and materials scientists, cracks of less than the critical length do manage to extend themselves, as we shall see in Chapter 15. However, the great point is that they generally do this so slowly that there should be plenty of time to spot them and do something about the situation.
Unfortunately things do not always work out quite that way. Professor J. F. C. Conn, who was Professor of Naval Architecture at Glasgow until recently, told me the story of a cook in a big freighter who was a little startled, when he went into his galley one morning to cook the breakfast, to find a large crack in the middle of the floor.
The cook sent for the Chief Steward, who came and looked at the crack and sent for the Chief Officer. The Chief Officer came and looked at the crack and sent for the Captain. The Captain came and looked at the crack and said ‘Oh, that will be all right -and now can I have my breakfast?’
The cook, however, was of a scientific turn of mind, and, when he had disposed of breakfast, he got some paint and marked the end of the crack and painted the date against the mark. Next time the ship went through some bad weather the crack extended a few inches and the cook painted in a new mark and a new date. Being a conscientious man he did this several times.
When the ship eventually broke in two, the half which was salvaged and towed into port happened to be the side on which the cook had painted the dates, and this, Professor Conn told me, constitutes the best and most reliable record we have of the progress of a large crack of sub-critical length,
‘Mild’ steel and ‘high tensile ‘ steel
When a structure fails or seems in danger of failing the natural instinct of the engineer may be to specify the use of a ‘stronger’ material: in the case of steel, what is known as a ‘higher tensile’ steel. With large structures this is generally a mistake, for it is clear that most of the strength, even of mild steel, is not really being used. This is because, as we have seen, the failure of a structure may be controlled, not by the strength, but by the brittleness of the material.
Although the measured value of the work of fracture does depend on the way in which the test is done, and it is difficult to get consistent figures, yet the toughness of most metals is undoubtedly reduced very greatly as the tensile strength increases. Figure 13 shows the sort of relationship which exists in simple carbon steels at room temperature.
It is quite easy, and not very expensive, to double the strength of mild steel by increasing the carbon content. If we do so, however, we may reduce the work of fracture by a factor of something like fifteen. In this case the critical crack length will be reduced in the same proportion – i.e. from 1 metre to 6 centimetres – at the same stress. However, if we double the working stress, which is presumably the object of the exercise, the critical crack length will be reduced by a factor of 15 × 22 = 60. So, if a safe crack was originally 1 metre long, it will now measure 1·5 centimetres – which would be thoroughly dangerous in a large structure.
Figure 13. The approximate relationship between tensile strength and work of fracture for some plain carbon steels. (By courtesy of Professor W. D. Biggs.)
With small components like bolts and crankshafts the situation is different, and it is meaningless to design for a crack a metre long. If we settle for an allowable crack length of, say, 1 centimetre, such a crack may be safe up to a stress of nearly 40,000 p.s.i. (280 MN/m2), and so there is a good case for using a high tensile material. Thus, one consequence of Griffith is that, on the whole, we can use high strength metals and high working stresses more safely in small structures than in large ones. The larger the structure the lower the stress which may have to be accepted in the interests of safety. This is one of the factors which tend to place a limit on the size of large ships and bridges.
The relationship between work of fracture and tensile strength which is sketched in Figure 13 is roughly true for simple commercial carbon steels. It is possible to get rather better combinations of strength and toughness by using ‘alloy steels’, that is, steels alloyed with elements other than carbon, but these are generally too expensive for large-scale construction. It is for these reasons that something like 98 per cent of all the steel which is made is ‘mild steel’, that is to say, a soft or ductile metal with a tensile strength of around 60,000 to 70,000 p.s.i. or 450 MN/m2.
On the brittleness of bones
Children, you are very little,
And your bones are very brittle;
If you would grow great and stately,
You must try to walk sedately.
R. L. Stevenson, A Child’s Garden of Verses
But, of course, the bones of children are not very brittle,* and Stevenson was writing rather charming nonsense. In the embryo, bones begin as collagen, or gristle, which is strong and tough but not very stiff (Young’s modulus about 600 MN/m2). As the foetus develops, the collagen is reinforced by fine inorganic fibres called osteones. These are formed chiefly from lime and phosphorus and have a chemical formula which approximates to 3Ca3(PO4)2. Ca(OH)2. In the fully reinforced bone the Young’s modulus is increased about thirtyfold to a value of about 20,000 MN/m2. However, our bones do not become fully calcified until some considerable time after birth. Naturally, young children are mechanically vulnerable, but on the whole they tend to bounce rather than break, as one can see on any ski-slope.
However, all bones are relatively brittle compared with soft tissues, and their work of fracture seems to be less than that of wood. This brittleness limits the structural risks which a large animal can accept. As we have already pointed out in connection with ships and machinery, the length of the critical Griffith crack is an absolute, not a relative distance. That is to say, it is just the same for a mouse as it is for an elephant. Furthermore the strength and stiffness of bone are much the same in all animals. This being so, it rather looks as if the largest size of animal which can be regarded as moderately safe is somewhere round about the size of a man or a lion. A mouse or a cat or a reasonably fit man can jump off a table with impunity; it is distinctly doubtful if an elephant could. In fact, elephants have to be very careful; one seldom sees them gambolling or jumping over fences like lambs or dogs. Really large animals, like whales, stick pretty consistently to the sea. Horses seem to present an interesting case. Presumably the original small wild horses did not very often break their bones, but now that man has bred horses big enough to carry him without tiring, the wretched creatures always seem to be breaking their legs.
It is well known that old people are particularly liable to break their bones, and this is generally attributed to a progressive em-brittlement of bone with age. No doubt this embrittlement does play some part in causing t
hese fractures, but it does not seem as if it were always the most important factor. As far as I know, there are no reliable data on the change of work of fracture of bone with age, but, since the tensile strength is only reduced by about 22 per cent between the ages of twenty-five and seventy-five, it does not look as if there were a very dramatic reduction. Professor J. P. Paul, of the University of Strathclyde, tells me that his researches seem to indicate that a more important cause of fracture in old people is the progressive loss of nervous control over the tensions in the muscles. A sudden alarm may cause a muscular contraction which is enough to break off the neck of the femur, for instance, without the patient having experienced any external blow. When this happens the patient naturally falls to the ground -perhaps on top of some obstacle-so that the fracture is blamed, wrongly, on the fall rather than on the muscular spasm. It is said that similar fracture can occur in the hind leg of certain African deer when they are startled by a lion.
* * *
* 1 Joule (1J) = 107 ergs = 0.734 foot-pound = 0.239 calories. Note that one Joule is roughly the energy with which an ordinary apple would hit the floor if it fell off an ordinary table.
*Since the oxygen consumption of the body is said to be higher during down-hill ski-ing than in any other human activity, much energy must also be got rid of in the muscles. However, most of the energy absorbed by the muscles is irrecoverable, and so the elastic strain energy storage of the tendons is no doubt to be preferred.
* Figures 2 and 4 are, of course, schematic. Generally the force-draw diagram will not be a straight line; but the same principle applies.
* On the other hand the rate of shooting of a cross-bow cannot match that of a hand-bow. The English longbow, for instance, could discharge up to fourteen arrows a minute and thus, when used en masse, could put up a very formidable cloud or barrage of missiles. It is calculated that about six million arrows were shot at Agincourt.