Why Things Break

A Hundred Times Too Weak

Pull harder, and in the end the bonds must let go. A bond gives out once it is stretched by about a tenth to a fifth of its length, so a solid in which every bond held to the last would break at about a tenth of its stiffness. Worked out properly, from the quantum mechanics of its atoms, iron should hold about 1,300 kilograms on each square millimeter, and glass about as much. That is the strength the bonds allow.

Real things fall far short of it. Ordinary structural steel stretches for good at about 25 kilograms on a square millimeter and breaks at 40 to 55, thirty to fifty times below what its bonds allow. Window glass is rated for about 4.5, a few hundred times below. A crystal of pure, softened copper starts to slip under a tenth of a kilogram on a square millimeter, some two thousand times below what its bonds should hold against sliding.

Something is doing the breaking for the bonds, and there turn out to be two different somethings. For glass, it is cracks. For metals, it is slip: a crystal does not break its bonds all at once, it slips them a row at a time.

Where the Pull Gathers

A load runs through a solid much as water runs down a channel, and anything in its way makes it crowd round. At the sides of a small round hole in a stretched plate the stress is three times the plate’s, whatever the hole’s size. An oval hole, long and narrow across the pull, gathers far more: at its ends the stress is the plate’s times one plus twice its length over its width. A crack is an oval squeezed nearly flat, and at its tip the stress has no limit but the size of the atoms.

It can be seen. Some clear plastics, strained, slow light polarized along the strain differently from light polarized across it. Between two crossed polarizing filters, of the kind in polarized sunglasses, a strained piece of such plastic shows bands of color, each band marking one level of stress, so that the bands crowd where the stress climbs. Engineers have long used it to see where parts are strained.

The plate in the scene has a round hole and a crack as wide, side by side under the same pull. Round the hole the bands stay broad. At the crack’s tips they close in, ring inside ring, as far down as anyone can look. A crack a couple of thousandths of a millimeter long, as sharp as an atom at its tip, multiplies the stress there about a hundred times. So the bonds at the tip reach their limit while the plate as a whole is nowhere near it. They part, the crack moves on a step, its tip as sharp as before and the crack longer, which makes it worse, and the break runs.

A Crack Must Pay Its Way

The rule for an oval hides a puzzle. If a crack’s tip gathers stress without limit, a sharp crack should break a plate under any load at all, yet cracked plates hold loads every day. What is missing is energy. A crack can grow only if growing pays. A stretched plate stores energy, as a stretched spring does, and a crack that lengthens lets the plate beside it relax and give some of that energy back. Making the crack’s two new faces costs energy, as making any fresh surface does.

A short crack frees too little to pay for its new faces. Past a certain length, at a given load, it frees more than enough, and runs. The longer the crack, the lower the load that sets it running: at half the load, a crack four times as long is needed. This was tested on thin glass bulbs and tubes, each scratched with a diamond, gently heated to ease the glass, and burst by pressure from inside. The bursting stress times the square root of the crack’s length came out very nearly the same for every crack.

Glass drawn into fibers told more. Rods a millimeter thick broke at about 17 kilograms on a square millimeter, and the finest fibers, three or four thousandths of a millimeter across, at about 350, twenty times as much. It looked as if the fine fibers had no room for big cracks. Later work found that their thinness was not the point. Fibers of any thickness, drawn and kept with their surfaces untouched, are about equally strong; what weakens glass is damage to its surface, from handling, from dust, even from the water in the air. An untouched fiber of pure silica glass holds about 600 kilograms on a square millimeter in air, and about 1,300 in liquid nitrogen, where water cannot attack it: close to what its bonds allow. The strength of glass is the strength of its surface.

A glass cutter does not cut. Its little wheel drives a line of crack into the surface, and a bend that stretches that face opens the crack right through the sheet. Cracks also explain why stone and concrete take pushing and not pulling. A squeeze presses a crack’s faces together and gives it nothing to open, while a pull opens it. Concrete holds about ten times as much squeezing as pulling, granite about twenty times, and so concrete that must carry a pull has steel bars set in it.

A Row at a Time

Metals hold a different surprise. A copper wire, a paperclip or a steel spoon bends and stays bent, where glass snaps. To slide one plane of atoms in a crystal over the next all at once, every bond across the plane must be stretched to its limit together, yet a crystal of pure copper starts to slip at about a two-thousandth of the stress that would take.

The answer is that a real crystal holds misfits called dislocations, places where one more row of atoms sits above a plane of the crystal than below it. Pushed sideways, a misfit moves along. A few bonds at its core break and remake one place on, and where it has passed, the half of the crystal above has slid one atom’s spacing over the half below. Moving a ruck along a carpet shifts a heavy rug in the same way.

It can be watched in a raft of soap bubbles. Bubbles of one size, a millimeter or two across, blown under soapy water, rise and pack into a crystal one bubble thick. Sheared between two coiled springs, the raft leans, and then slip begins at one end with a dislocation, which runs along a row and out at the far end, taking seconds to cross.

In a metal, dislocations can be seen moving under an electron microscope, in a thin foil of aluminum. They are everywhere. A softened metal holds hundreds of kilometers of them in every cubic centimeter, and a hard-worked one around a million. Bending a paperclip moves countless dislocations, and since nothing moves them back, it stays bent. They also multiply as they move and tangle in each other’s way, which is why bent metal grows harder. Alloys add foreign atoms and fine particles that pin them. Most of metallurgy is the art of making dislocations hard, but not too hard, to move.

Glass has no crystal and no rows to slip, so at room temperature a pane can only spring back or break. That also ends the old story that window glass flows downward over the centuries. Worked out for the glass of medieval windows, the flow comes to about a millionth of a millimeter in a billion years. Old panes thicker at the bottom were made uneven, and were often set with the thick edge down.

Bend or Snap

Dislocations also decide whether a cracked part bends or snaps. At a crack’s tip in a metal, the crystal slips before its bonds can split. The tip blunts, and to move on, the crack must drag a zone of bent metal along with it, which takes thousands of times more energy than the bare new faces a crack in glass makes. That energy is a material’s toughness, and it is a different thing from strength. Flawless glass is strong but not tough. Mild steel is both.

Mostly. In the iron crystals of ordinary steel, dislocations need help from the jiggling of the atoms to get from one place to the next. Cold, they hardly move, the crack’s tip cannot blunt, and the steel splits straight through its crystals along flat planes of atoms, almost as glass breaks. Metals whose atoms pack the other way, such as copper, aluminum and the stainless steel of kitchen sinks, stay tough however cold they get. Tanks for liquefied gas at 160 degrees below zero are made of them, or of steel with nine percent nickel, in which the change comes far colder still.

The test for it is a pendulum, the Charpy test, which steelmakers still run. A bar ten millimeters square, with a notch cut across one face, lies against two anvils, and a heavy hammer swings down and strikes it behind the notch at about five meters a second. How high the hammer swings on after the blow shows how much energy the bar took. Warm, a structural steel folds round the hammer and takes much of its swing. Chilled well below freezing, the same steel snaps, and the hammer swings on almost as high as it started. The broken faces tell the same story: a tough break is dull and torn, a brittle one bright and granular, glittering with the flat faces where its crystals split.

In the Second World War, American yards built about five thousand welded merchant ships at great speed. Of the 4,694 that a Navy board studied, 970 had cracked somewhere, 127 of them seriously, and eight had broken in two. The most famous to break in two was a new tanker, the Schenectady, at her fitting-out pier one January night. The water was still, the crack was heard a mile away, and only her bottom plating held. The air was a few degrees below freezing and the water four degrees above it, and the stress in her deck was about seven kilograms on a square millimeter, a small fraction of what her steel could carry.

The board found the fractures “caused by notches and by steel which was notch sensitive at operating temperatures”. Every fracture it examined began at a notch: a square corner of a hatch, a cut in the edge of a deck, a flawed weld. The Schenectady’s began at a defective weld where a plate of her superstructure met the top edge of her side. The steel was ordinary ship plate, and tests showed that it turned brittle at temperatures above freezing. The hulls were also welded into one piece. In a riveted hull a crack usually stops at the next seam, while in a welded one it can run right round the ship. Plates cut from the broken ships told the rest: those in which cracks had started took only about ten joules in the Charpy test at the temperature they broke at, and those that stopped cracks about twice as much.

The fixes worked. Hatch corners were rounded and strengthened, and riveted steel straps were added as crack stoppers; by the board’s count, no crack ever got past one. Ship steel has since been tested for toughness at the temperatures ships sail in, and ships today carry bands of extra-tough steel along their decks and sides to stop a running crack.

A Little Each Time

A load a part carries easily once can still break it if it comes again and again. Bend a paperclip once and it stays bent. Bend it back and forth and it usually breaks within ten to thirty bends, warm where it bends. Each bend drives dislocations back and forth. Under a microscope, the bands where iron slips can be seen widening, reversal after reversal, into cracks, even where there is no flaw to start one. Each load after that drives the crack a little further. The breaking is called fatigue.

It became a matter of life and death on the railways. In one early crash a locomotive’s axle broke, and the wreck of the train behind it killed at least 55 people. Axles were breaking after years of service, from a fine crack creeping round the shoulder where an axle narrowed to its bearing, and a generous curve there made them last. Test axles kept turning under load showed the rule. A stress well below the breaking strength will break an axle if it comes often enough, and it is the size of the swing in stress that matters most. Below a certain swing, the steel tested did not break in any number of turns that could be reached. Aluminum has no such floor.

The plate below shows it in polarized light, its cracks growing as they would in a metal part, each pull drawn standing for thousands. Each is far below what would break the plate at once. But at the hole’s edges, where the pull is three times the plate’s, cracks start, and each pull drives them a little further. The longer they grow, the more the pull gathers at their tips and the faster they go, until they are long enough to run, and the next pull breaks the plate.

The first jet airliner, the de Havilland Comet, met fatigue. Two broke up in the air within three months, killing 56 people. To find out why, engineers sat a whole Comet in a tank of water and pressed its cabin up and down, as every flight does. After the equivalent of 3,060 flights its skin split from the corner of an opening. Near the corners of its windows and other openings the stress had been half as much again as the designers had worked out, enough for a crack from a bolt hole to grow through the skin in about a thousand flights. The windows are often said to have been square. Their corners were rounded, though not enough, and the cracks began at an opening in the roof and at an escape hatch.

Since then aircraft have been built to fly on safely with a crack in them, and inspected often enough to find one long before it can reach the length at which it runs. A fatigue break also keeps its own record. Its face is smooth where the crack crept, often ringed with marks where the loading changed or paused, and rough where the last of the metal tore.

Living With Cracks

Nothing large is free of flaws, so much of the craft of making strong things is the craft of handling cracks. One way is to keep a surface squeezed. Toughened glass is heated and its faces chilled with jets of air, so that they set first and are left squeezed as the inside cools and shrinks. A crack in the surface must overcome the squeeze before it can open, and the glass is about four times as strong. Prince Rupert’s drops, beads of molten glass dripped into water, take this to the extreme. Their heads survive a hammer, their skin squeezed, as measured in polarized light, by forty to seventy kilograms on a square millimeter. Snap the thin tail, though, and a crack reaches the stretched core and races through the whole drop at one and a half to two kilometers a second, turning it to powder.

Another way is to stop cracks. A round hole drilled at a crack’s tip blunts it. The pull still gathers at the hole’s edge, but far less than at a tip as sharp as an atom, and the crack often stops there. It is a common repair for fatigue cracks in steel bridges. Fibers set in a resin, or the riveted straps on the wartime ships, give a running crack an edge to stop at. A third way is to find cracks before they matter: in aircraft and bridges, the time between inspections is set by how fast a crack can grow.

Size works against large things twice. The first way is weight. Strength grows with area and weight with volume, which limits how big a cord, a bone or an animal can be. The second way is flaws. A larger piece of a brittle material has more room for a bad flaw, and it is only as strong as its worst one, so large pieces are weaker on average than small ones. Small things win both ways, which is part of why an insect survives a fall from any height: the air slows a small body to a gentle speed, and its legs are strong for its weight.

How Strong Could Anything Be?

Made small enough to be free of flaws, materials come close to the strength of their bonds. Whiskers of iron a few thousandths of a millimeter thick, almost perfect single crystals, have held about 1,300 kilograms on a square millimeter. Pressed with a diamond-tipped probe, a single sheet of carbon atoms, graphene, broke at about 13,000, among the highest strengths ever measured. Building with such things is another matter. A long cable has a long length in which to hold its worst defect, and the strongest carbon fibers made today reach about 800, a sixteenth of graphene’s strength. Carrying the strength of a perfect sheet up to the size of a cable is still unsolved.

Why does a solid push back at all when it is squeezed? A single atom keeps its size because squeezing an electron’s cloud tighter round its nucleus raises the electron’s energy of motion faster than it lowers its electric energy. A lump of matter needs more than that. It has been proved that a lump of electrons and nuclei has an energy in proportion to the number of its particles, so that two stones hold twice the energy of one. The proof needs Pauli’s exclusion principle, that no two electrons can share a state, and cannot do without it: without the rule, matter would pull itself in tighter the more of it there was. The same rule is why atoms push back hard when their filled layers of electron clouds are pressed together. The electrons are forced into patterns that cost more energy the more the clouds overlap, and the push that results is electric.

What is still out of reach is working out a material’s toughness, or how many loads it will survive, from its atoms alone. The pieces are known: bonds, dislocations, cracks. But a crack’s tip spans every scale at once, from single bonds parting to centimeters of bent metal, and no calculation yet carries the one into the other. Even why steel turns brittle in the cold is being argued again. A recent study of the fracture surfaces of brittle iron, dimpled on the scale of millionths of a millimeter, argues that even a brittle crack moves by slipping a little at a time. Glass keeps a puzzle of its own. A crack in it could in theory run almost as fast as sound travels along its surface, yet it tops out at about half that and branches as it goes, and only in recent years have computers begun to show why.