The New Science of Strong Materials by J. E. Gordon

There is a crack in everything…

If you’re going to read J. E. Gordon’s The New Science of Strong Materials, you had better like reading about cracks. You don’t have to know mathematics: Equations appear in only a few locations in the main text, and you can skip them without losing the plot. But materials scientists are obsessed with cracks. They spend whole careers studying the intricacies of how cracks propagate. They dream about cracks, or more often, have nightmares. Normal people might stare at a crack on the ceiling because the conversation has taken an awkward turn. A materials scientist will stare at it because it’s a crack and cracks are interesting.

Here’s one interesting fact about cracks: Take a rod made of glass, with about the same dimensions as a broom handle. Now try to bend it to a 90 degree angle. You will not succeed. Getting the rod to bend even a tiny amount requires a large amount of force. Even if you can supply that force, the rod will shatter into pieces long before it bends by a full 90 degrees. You’d think the story would be the same for a shrunk down version of the same rod. After all, the thing is in the same shape and made of the same material, just smaller, so all the strains in the material should be the same. But the tiny rod can be bent by 90 degrees. And it does not deform permanently, like a metal might. Let go of it, and it will snap right back to being perfectly straight.

The secret of the trick is that most glass objects are imperfect, with surfaces being covered by tiny cracks. The surface will pick up microscopic damage very quickly when other objects come into contact with it, even if you think you’re handling it very gently. So it’s nearly impossible for a large piece of glass to be free of cracks. But if it were, it would be astonishingly strong. The theoretical limit on the strength of glass as determined by the strength of the bonds holding it together is rather high, it’s just that experimentally the overall material turns out to be weaker than the sum of its bonds. The very thin glass fiber in the experiment can be made and handled without forming surface cracks and as a result it retains most of that strength and can be bent by absurd amounts. But how could such tiny imperfections make such a huge difference to the strength of the material?

Actually, what makes things strong or weak in the first place? What do we even mean by strength?

The goal of having strong materials is so that they can exert forces. A railway bridge needs to exert upwards force on all the trains going over it so they don’t fall. In general, we ask materials to exert force in one of 3 ways:

1. Tension (trying to pull the material apart).
2. Compression (trying to crush/squeeze the material)
3. Shear (trying to slide the layers of the material past each other (This is the one that lets beams transfer a vertical force over a horizontal distance: You can be standing in the middle of a beam, while the beam may only be supported on either end. The vertical force of your weight is transferred horizontally to the supports at the ends of the beam. (There’s a cost for this horizontal transfer of vertical force: The top of the beam is in compression, and the bottom of the beam in tension. I.e. the beam wants to bend. These compression and tension forces scale with the horizontal distance across which the beam needs to transfer vertical force, so long beams need to be made of materials that are strong in both tension and compression.)))

There is a fundamental rule here that applies to all 3 cases: No material can exert a force without being stretched, squeezed, bent or otherwise deformed at least a little. Even if a single feather floats down and comes to rest on top of a solid block of steel, the only way that the steel is able to support the weight of the feather is that it is dented by a microscopically tiny amount.

The way to understand this is to understand that a solid material is made up from atoms, and those atoms are all bonded together. Chemists and quantum physicists would have a great many complicated things to say about those bonds, but for us, all we need to know is that they act pretty much like tiny little “electric springs”. At their natural length, these springs don’t exert any force, but if you squeeze them to less than their natural length, they push back outward. If you stretch them to longer than their natural length, they pull back inward. And if you stretch them really really far, then they break. Deform a solid, and the bonds are stretched and squeezed away from their natural lengths, and they pull and push to get back to how they were. The greater the deformation of the solid, the greater the force. That’s where the force comes from that holds up bridges, keeps the wings from coming off of airplanes, and so on.

Some materials are stiffer than others. A stiff material is one that will exert a very great force after being bent only a tiny little bit. When bent by an amount that is not too extreme, the opposing force a material supplies is proportional to the amount it is bent, so we can take the ratio between force and amount of bending to get a measure of the stiffness of the material, which is called the Young’s modulus. Diamond is a very stiff material, and has a correspondingly very high Young’s modulus of 170 million psi. Steel is a bit less stiff at 30 million psi, wood still less at about 2 million psi, and rubber is really floppy with a Young’s modulus of about 1000 psi.

So are the stiffer materials the stronger ones? No! Strength is all about how hard you have to pull on something to break it. Stiffness is more about how little it moves when you’re pulling on it. Materials like ceramic are stiff, but not very strong. A ceramic mug breaks when it’s dropped on the ground, but during daily use it holds its shape very well: The amount it deforms when picked up by the handle is not noticeable. Contrast this with steel: Both steel and ceramic are stiff, with a high Young’s modulus. They also both have strong atomic bonds between their atoms. Yet a steel mug could be dropped on the floor with no problems at all. (No problems for the mug, that is. I’m not making any guarantees about the floor!) There’s no obvious reason why one of these materials should be brittle and one strong. The answer, of course, involves cracks.

Cracks tend to form when a material is under tension. Compression would just push cracks in the material closed again, which is why brittle materials work fine for making compression structures like castle walls, or arches. But tension is dangerous. And remember that tension shows up even in situations like load-bearing beams, where one side of the beam is under tension. Certainly the hulls of ships will frequently be under tension in various ways as they roll on the waves.

In 1903, the British Admiralty, presumably wanting to know if their ships were well designed, measured the strains in the hull of a steel ship at sea: the H.M.S. Wolf. After calculating the force that would be required to break the ship, based on the strength of steel, they concluded that the ship was built with a construction far stronger than needed to withstand the waves. Yet many other ships of similar design continued to break at sea. The reason why the test had failed was not uncovered until a decade later. The ship builders had accounted for holes in hull such as hatches, and portholes and so on, by subtracting out the missing material, and ensuring that what remained had enough total strength to take the stress. But in 1913, Professor Charles Inglis published a paper showing that merely subtracting the missing material from the calculation was wrong. The ship builders had been assuming that when a porthole is cut into the side of a ship, the stress is distributed evenly over the remaining parts of the steel hull. But this is not the case. The stress is in fact much more concentrated around the porthole, and this is where the material is most likely to fail. The test on the Wolf didn’t show any dangerous readings because none of the measurements were taken near holes in the hull.

Inglis’s calculations show that, even more than a hole, stress is very concentrated at the tip of a crack. If we’re unlucky, this high stress causes the material at the tip of the crack to break, lengthening the crack, and creating a new high-stress point where the material also breaks, lengthening the crack still further in an unhappy chain reaction. This is why bridges are regularly inspected for cracks, and even relatively small cracks are cause for concern.

We can figure out whether or not a crack will propagate using energy conservation. Doubling the length of a crack quadruples the amount of energy released, since not only is the crack twice as long, but the longer crack allows the material to flex more, and so it can move by twice the amount and thus release twice the energy per unit length. On the other hand, creating the crack uses up energy proportional to the length of the crack because it takes energy to separate the material from itself (this is called the work of fracture). At the very least, the work of fracture includes the energy needed to break the connecting bonds between the two pieces, though there can be other contributions too. The upshot is that short cracks will not go anywhere because it takes too much energy to overcome the work of fracture, but once a crack is long enough, it will generally grow to run straight through the material. The goal in a strong material then is to have a very large work of fracture, so that it takes a lot of energy for cracks to propagate. This means that the dangerous length for cracks in such a material is also very high, and we can be safe in not worrying about them. On the other hand, if the work of fracture is very low, then even microscopic cracks could be dangerous. That is the situation for brittle materials like glass and ceramic. It was the engineer Alan Griffith who extended Inglis’s ideas about stress concentrations to microscopic cracks, and came up with an equation for the critical dangerous crack length.

Metals tend to be very strong because they have a high work of fracture. This is down to the fact that (most) metals are ductile, which is down to dislocations. A dislocation is a place in a metal’s crystal structure where the crystal isn’t quite perfect, but there’s instead a defect where the crystal pattern isn’t followed perfectly. At a dislocation, atoms in a metal have a very easy and clever way of slipping past each other and changing which neighbors they’re bonded to. When a bit of steel (or aluminum, or most other metals) is strained too hard, it won’t break immediately. Instead, all the dislocations naturally present in the crystal will move around and the atoms will change partners and the metal will reshape itself like clay to try and accommodate the strain. Even if the metal does eventually fail, it will only fail after an enormous amount of energy has already been put into moving around numerous dislocations and repartnering countless atoms. All of that energy required just to separate two bits of metal from each other translates to a large work of fracture, which means that cracks in metals have to be rather long to be dangerous. That’s why steel and aluminum are such great engineering materials. Ceramic is brittle because it doesn’t have mobile dislocations like metals do, and so its work of fracture is much less.

Some other materials have different ways of stopping cracks. If we deliberately make a material of alternating strong layers and weak layers, then a crack heading perpendicular to the layers will be persuaded to turn sideways when it encounters a weak layer, but will soon run out of steam and stop propagating sideways too, since the tension is in the wrong axis to sustain it. Weakening some layers in the material oddly makes it stronger. Wood is strong in a similar way, being made from many strong fibers weakly bound together.

There’s loads of other good stuff in this book. Gordon relates an amusing story of a chemist he worked with who devised a material containing LOTS AND LOTS OF REALLY STRONG BONDS. Surely this would end up being the strongest material ever? In fact, it turned out to be incredibly brittle and weak. That chemist had neglected the importance of having a high work of fracture to stop cracks, and the strong bonds were of minimal help there.

Gordon also discusses the use of the wood-framed Mosquito airplane in World War 2. (Wooden planes are great, as long as you remember to sand the plywood before gluing it.) There’s also an excellent chapter on the history of iron and steel production and how high quality steel has become so miraculously cheap in modern times.

Besides telling us a great many fascinating facts about how materials work, this book has something interesting to say about science in general. Materials science started not from a place of zero-knowledge, but rather started after people had already been working with materials and using them to build things for thousands of years. So naturally there was a large amount of built-up tradition around how to do certain things, and a large amount of built-up cultural belief in how things worked. Probably this was mostly fine and correct and got decent-enough results. But Gordon points out that a surprising amount of the time, the traditional knowledge was either useless, or was actively unhelpful “anti-knowledge”. Lots of ancient societies made animal or sometimes human sacrifices to imbue their structures with strength, builders of wooden ships made designs that leaked when they needn’t have, due to their lack of engineering knowledge. It took all the way until 1913 for Inglis to come along and publish his ideas about stress concentration, and even then the practical engineers initially dismissed them. Everyone but the physicists hated the idea of dislocations being responsible for ductility in metals. So both the academics and the practicing craftsmen were guilty of this kind of dogmatic thinking. In a way, one can’t really blame them. Materials are something we all have everyday experience with. Introducing all these invisible peculiarities that were supposed to be important – dislocations, stress concentrations, microscopic cracks – surely must have felt unnatural.

I’m not sure if we should take anything away from this other than “knowing things is hard” or “even really obvious-sounding questions can have very complicated answers.” Perhaps “trying to figure things out scientifically using experiments and algebra performs surprisingly well relative to relying on tradition”. But I guess we have learned one thing for sure: “For a material to be strong, it must not crack easily.”