by J E Gordon
When we do the mathematics it turns out that the only sort of elasticity which is completely stable under fluid pressures at high strains is that which is represented by Figure 5. With minor variations, this shape of stress-strain curve is very common indeed for animal tissue and particularly for membranes. You can feel that this is so if you pull on the lobe of your ear.
Figure 5. Stress-strain curve for typical animal tissue.
It will be noticed that Figure 5 seems to beg the question of whether the stress-strain curve for such materials really passes through the origin (the point of zero stress and strain) or whether there is still a finite tension in the material when there is no strain – a state of affairs which is no doubt calculated to shock the souls of engineers brought up on Hookean materials like steel. As far as one can see, however, in the living body there seems to be nothing really corresponding to an ‘origin’: that is, there is, apparently, no real position of zero stress and strain (as there would be in any structure made from, say, soap-films). The arteries, at any rate, are permanently in tension within the body, and, if they are dissected out of a living or a freshly dead animal, they will shorten fairly noticeably.
As we shall see in the next section, this tension is perhaps an additional device to counteract any tendency of the artery to alter in length as the blood-pressure changes, or else it may represent a belated attempt to equalize the longitudinal and circumferential stresses within the artery wall – in other words an attempt to get back to the conditions of surface tension which may have existed in the dim past. When people are subjected to severe and prolonged vibration – for instance in the case of foresters using chain saws – this tension may be lost and the arteries elongate and take up a meandering, convoluted or zig-zag path.
Poissorfs ratio – or how our arteries work
The heart is, in effect, a reciprocating pump which discharges blood into the arteries in a series of fairly sharp pulses. The work of the heart is eased, and the general well-being of the body is served, by the fact that, on the pumping or systolic part of the cardiac cycle, much of the excess of high-pressure blood is accommodated by the elastic expansion of the aorta and of the larger arteries; this has the effect of smoothing the fluctuations in pressure and generally facilitating the circulation of the blood. In fact the elasticity of the arteries does much the same job as the air-bottle affair which engineer* often attach to mechanical reciprocating pumps. In this simple device the surge of pressure which accompanies the discharge stroke of the pump piston is smoothed out by arranging for the liquid which is being pumped temporarily to compress a supply of air which is trapped in a suitable bulb or container. When the valve of the pump shuts at the end of its stroke (as that of the heart does in diastole), the liquid continues to be driven on its way by the recovery and expansion of the trapped air (Figure 6).
Figure 6. The elastic expansion of the aorta and the arteries performs the same function in smoothing the fluctuations of blood-pressure as does the air-chamber attached to an engineer’s reciprocating pump.
This rhythmic expansion and relaxation of the artery is necessary and beneficial; and in fact, if the artery walls stiffen and harden with age, the blood-pressure is likely to rise and the heart has to do more work, which may not be good for it. Most of us know about this, but not many people stop to consider what happens about the strains in the artery walls.
As we calculated in Chapter 6, the longitudinal stress in a cylindrical vessel, such as an artery wall, is just half the circumferential stress; this will always be the case, whatever the walls of the container are made of. Therefore, if Hooke’s law were directly and crudely obeyed, the longitudinal strain would also be half the circumferential strain, and the total extensions would be in due proportion, taking the dimensions into account. Now a major artery – such as those which supply blood to our legs – might be something like a centimetre in diameter and perhaps a metre long. If the strains were really in the ratio of two to one, a simple calculation shows that a change of diameter of half a millimetre -which is easily accommodated within the body – would be associated with a total change of length in the artery of about 25 millimetres or about an inch.
It is obvious that a change of length of this magnitude, occurring seventy times a minute, cannot and does not take place. If this sort of thing really happened, our bodies just would not work. To take an extreme case, one has only to imagine such a thing happening in the blood-vessels which serve the brain.
Fortunately, in real life, the lengthwise strains and extensions in pressurized tubes of all sorts and kinds are very much less than one might have anticipated or feared from this rather too simple argument. That this is so is due to something called ‘Poisson’s ratio’.
If you stretch a rubber band it gets very noticeably thinner; and much the same thing happens with all solids, although with most materials the effect is less conspicuous. Contrariwise, if you shorten a material by compressing it, it will bulge out sideways. Both of these are elastic effects and they disappear when the loads are taken off.
The reason why we do not notice these lateral movements in things like steel and bone is that both the longitudinal and the transverse strains are so small; but the effects are there all the same. The fact that this happens in all solids and that such behaviour is significant in practical elasticity was first observed by the Frenchman S. D. Poisson (1781-1840). Although he was born into rather dramatic poverty and got very little formal education before he was fifteen, Poisson was made an Academician – one of the highest honours France has to offer – at the age of thirty-one for his work on elasticity.
As we said in Chapter 3, Hooke’s law says that
Thus, if we apply a tensile stress, s1, to a flat plate, the material will be elongated or stretched elastically so that there will be a tensile strain in the direction in which we are pulling of
However, the material will also be contracted sideways (i.e. at right angles to s1) by some other strain which we may call e2. Poisson found that, for any given material, the ratio of e2 to e1 is constant, and this ratio is what we now call ‘Poisson’s ratio‘. We shall use the symbol q in this book. Thus, for a given material subject to a simple uniaxial tension stress s1
e1 the strain in the direction of s1 is often called the ‘primary strain’; the strain caused by s1 at right angles to itself, so to speak, is called the’ secondary strain’ (Figure 7).
Figure 7. When a solid is stretched by a tensile stress s1 it extends in the direction of s1 by a primary strain el but it also contracts laterally by a secondary strain e2.
From what we have just been saying,
and since e1 = s1/E (which is Hooke’s law)
then
Thus if we know q and E we can calculate both the primary and the secondary strains.
For engineering materials like metals and stone and concrete, q nearly always lies between ¼ and ⅓. The values for Poisson’s ratio for biological solids are generally higher than this and are often around ½. Teachers of elementary elasticity will tell you that Poisson’s ratio cannot have a higher value than ½ – otherwise various naughty and inadmissible things would happen. This is only partly true, and the values for some biological materials can sometimes be very high indeed, often well over unity.* The experimental value for the Poisson’s ratio for my tummy, measured recently by me in my bath, is about 1 -0 (see the footnote on p. 162).
Thus, as we have said, the effect of Poisson’s ratio is that, if we pull upon a piece of material, such as a membrane or an artery wall, in one direction it will get longer in that direction, but it will contract, or get shorter, in the direction at right angles. So if two tensions are applied, at right angles to each other, the effects will be additive and the strains will be less than we should expect if either of the stresses were applied separately.
For two simultaneous stresses, s1 and s2, the total strain in the direction of s1 will be
and the total strain in the direction of s2 will be
>
Harking back to Chapter 6, the consequence of the existence of Poisson’s ratio is that the longitudinal or lengthwise strain in the wall of a tubular pressure vessel which obeys Hooke’s law is
where r = radius, p = pressure, and t = wall thickness.
It follows that the longitudinal elastic extension of a tube is much less than one might expect; for a Hookean material with a Poisson’s ratio of i there will be no movement at all. In fact, as we have seen, the artery walls do not obey Hooke’s law, and it is also probable that their Poisson’s ratio is higher than i; possibly these two effects offset each other, because experimentally very little lengthwise movement is observed.* No doubt the fact that arteries are permanently stretched within the body is a precaution against any residual longitudinal strain.
The effects of Poisson’s ratio are probably of very great importance in animal tissues; but they are also significant in engineering and the matter is continually cropping up in all sorts of connections.
It should perhaps be added that, whereas the aorta and the principal arteries expand and contract elastically with each beat of the heart, in the manner we have just been discussing, the state of affairs with the smaller arteries is usually rather different. The walls of these lesser vessels are provided with muscular tissue which can increase their effective stiffness and so, by restricting the diameter, is able to control the amount of blood which is able to pass to any particular region of the body. In this way the local distribution of the blood-supply is adjusted.
Safety – or the toughness of animals
Animals quite often break their bones and they sometimes tear their tendons, neither of which have the sort of elasticity we have been discussing; but it is very noticeable that the mechanical fracture of soft tissues seems to be rare. There are several reasons for this. Being so soft, skin and flesh can sometimes evade the effects of a blow by deflecting and escape with a bruise. The question of stress concentrations, however, seems to be more interesting, since the majority of soft animal tissues appear to be almost immune from this major cause of engineering catastrophes. For this reason the need for a factor of safety is much reduced, and thus the structural efficiency, that is, the load which the structure carries in proportion to its weight, may be quite high.
This immunity is not just a matter of being soft and having a low Young’s modulus. Rubber is indeed soft and has quite a low modulus, yet many of us can remember, as children, having taken our blown-up rubber balloons out into the garden where they very soon burst with a bang on encountering the prickles of the first rose-bush. As children, we did not realize that, owing to the stress concentration and to the 10w work of fracture of rubber, a crack spreads very rapidly front a pin-hole in stretched rubber, and it is rather doubtful if our tears would have been much abated if we had. However, the membrane of a bat’s wing, for example, although it is much stretched in flight, does not seem to behave in this way. If the wing does get punctured, the tear seldom spreads and the injury soon heals, although the bat may be using its wings continually.
The explanation lies, I think, in the very different elasticities and works of fracture of rubber and of animal membranes. There are at present virtually no data available about the works of fracture of biological soft tissues; but the shapes of the stress-strain curves are in most cases pretty well known, and this latter factor does seem to have a big influence upon the probability of fracture.
The shell-membrane of an egg seems to afford an interesting example – this is the membrane which you encounter at breakfast, just inside the shell of your boiled egg. It is one of the few biological membranes which obey Hooke’s law, in this case up to its breaking strain of about 24 per cent* A simple but slightly messy experiment with a raw egg will show that egg membranes tear very easily. This is, of course, what they are there for, since the first thing the chick has to do is to get out of its egg, which it does by pecking with its beak. Incidentally the egg-shell itself, with its rounded domed shape, is difficult to break from the outside but easy to break from inside.
Egg membranes are rather exceptional, in that they exist in order to be broken after they have served their purpose of conserving the moisture in the egg and keeping out infection; as we have said, they possess a special sort of elasticity, very possibly for this reason. However, the great majority of soft tissues have an elasticity which is quite different and is very much like Figure 5; functionally, most of these tissues need to be tough. Although all the scientific reasons are not perfectly clear, it does seem that, pragmatically, materials with this type of stress-strain curve are extremely difficult to tear. One reason is, perhaps, that the strain energy stored under such a curve – and therefore available to propagate fracture (Chapter 5) is minimized.*
As we have said, a very high proportion of animal tissues behave, elastically, much in the manner of Figure 5.1 must confess that, when this information first dawned on me, it seemed to me to indicate an eccentricity or quirk on the part of Nature, who, poor thing, did not know any better, not having had the benefit of an engineering education. After a good deal of rather blundering research into the elementary mathematics of the problem it is now beginning to dawn on me that, if one has need of a structural system which will work reliably at really high strains, this is the only sort of elasticity which will serve. In fact, the achievement of this kind of stress-strain curve in animal materials represents a really essential condition for the evolution and continued existence of the higher forms of life. Biologists, please note.
The constitution of soft tissues
Perhaps partly for these reasons the molecular structure of animal tissue does not often resemble that of rubber or artificial plastics. Most of these natural materials are highly complex, and in many cases they are of a composite nature, with at least two components; that is to say, they have a continuous phase or matrix which is reinforced by means of strong fibres or filaments of another substance. In a good many animals this continuous phase or matrix contains a material called ‘elastin’, which has a very low modulus and a stress-strain curve something like Figure 8. In other words elastin is only about one stage removed, elastically, from a surface tension material. The elastin is, however, reinforced by an arrangement of bent and zig-zagged fibres of collagen (Plate 4), a protein, very much the same as tendon, which has a high modulus and a nearly Hookean behaviour. Because the reinforcing fibres are so much convoluted, when the material is in its resting or low-strain condition they contribute very little to its resistance to extension, and the initial elastic behaviour is pretty well that of the elastin. However, as the composite tissue stretches the collagen fibres begin to come taut; thus in the extended state the modulus of the material is that of the collagen, which more or less accounts for Figure 5.
Figure 8. Approximate stress-strain curves for elastin and collagen.
The role of the collagen fibres is not merely to stiffen the tissue at high strains; they also seem to contribute very much to its toughness. When living tissue is cut, either accidentally or sur gically, in the first stage of the healing process the collagen fibres are re-absorbed and disappear, temporarily, for a considerable distance around the wound. It is only after the gap has been filled and bridged by elastin that the collagen fibres are re-formed and the full strength of the tissue is restored. This process may take up to three or four weeks, and in the meantime the flesh around the wound has an almost negligibly low work of fracture. It is for this reason that, if a surgical wound has to be reopened within two or three weeks of the original operation, it may be dificult to get the new stitches to hold.
Figure 9. Hypothetical morphology of elastin.
(a) Resting or unextended state. Chain molecules folded or mainly folded within droplets.
(b) Extended state. Chain molecules pulled out of droplets.
Collagen exists in various forms, but it may consist of twisted strings or ropes of protein molecules, and its resistance to extension is basically due to the need to stretch the bo
nds between the atoms in the molecules: that is to say, it is a Hookean material much like nylon or steel. Why then does elastin behave as it does, almost like surface tension? The short answer is that nobody really knows, but Professors Weis-Fogh and Andersen have suggested that this behaviour may in fact be due to a modified form of surface tension. According to this hypothesis, elastin may consist of a network of flexible long-chain molecules operating within an emulsion. Since the molecules of the network are wetted by the droplets – but not by the substance between them – it is energetically preferable for most of the length of these molecules to remain coiled or folded up within the drops (Figure 9a). Under tension, they are dragged out of the drops and extended (Figure 9b).*
Much of our body consists, of course, of muscle, which is an active substance capable of contracting so as to produce the tensions which are needed in the tendons and elsewhere. Muscle, however, contains collagen fibres, which can only play a passive part elastically. When dead muscle is stretched it has a stress-strain curve which is, again, very much like Figure 5, and it seems possible that the function of the collagen in muscle is to limit the extension of the muscle when it is in its relaxed or extended state: in other words it acts as a sort of safety-stop.
As we have said, another purpose of collagen fibres in flesh is to put up the work of fracture. This is a good thing for the animal, but it is inconvenient for the people who want to eat its flesh. In other words, it is collagen which makes meat tough. Nature, however, does not seem to be on the side of the vegetarians, because she has arranged, in her wisdom, that collagen should break down to gelatin – a substance of low strength when wet – at a somewhat lower temperature than that which elastin or muscle can withstand. The process of cooking meat therefore consists in converting most of the collagen fibres into gelatin (which is jelly or glue) by roasting or frying or boiling. It is science of this kind which restores one’s faith in the beneficence of Providence.