Cracks - Part Two
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In ancient times, one way to split blocks of stone out of a quarry was to insert dry wood into cracks or rows of holes. Soaking the wood made it expand, causing the rock to break away. Cleaving slate into thin plates makes a very good roofing material. Another laminar mineral, mica, has been used for the windows of lamps, for the support of the heater in an electric iron, and for the insulator in a capacitor.
Imagine that you have created a work of art based on two big sheets of glass. It is shipped to an art gallery. When the crate is opened the glass is found to be cracked all over. What do you say? When this happened to a famous work, "La mariée mise à nu par les célébataires", or "The Large Glass", by Marcel Duchamp, the artist did not seem to be annoyed - he simply rebuilt it. At one bottom corner of the work, the cracks show beautifully how the stresses lay in the glass. The cracks run towards the corner, but then turn away and hit the sides at angles not far from ninety degrees. Other copies of the work do not have the cracks. In the middle of the glass we see two types of cracks - primary cracks radiating towards the edges, and secondary cracks, where the long shards have broken into small, roughly rectangular pieces.
The diagram below shows an idealised picture of cracks emanating from the centre of a pane of glass. The electric field of a charge in a square metal box would be similar.
Here is a piece of glass that has had some pieces chipped off. These shapes are called conchoidal fractures. Where flints occurred, some prehistoric people were skilled in breaking flakes off to leave extremely sharp edges for arrow heads and axe heads. Useful cracks indeed.
Can other cracks ever be useful? These are cracks in the glaze of a small Japanese cup. They are induced quite deliberately by using a glaze that shrinks much more than the ceramic on cooling. Some ceramics even have two layers with different average sizes of the cells, creating an interesting effect. Good examples of crackle may be found in Song Guan ware and Longquan ware from China. Excellent examples may be seen in the Baur Collection in Geneva, the Percival David and British Museum collections in London, and the Ashmolean Museum in Oxford.
The initial cracks may follow the stresses induced when the pot was thrown. Note how the cracks generally meet at angles that are close to 90 degrees. Suppose that one crack has already been created. As the glaze shrinks, there will be tensile stress in all directions. But tension cannot cross a crack, and so, near a crack, the tension must be almost parallel to the crack, causing new cracks to be at right angles. For the same reason, a crack will usually reach the edge of an object at about 90 degrees.
This type of effect can sometimes be seen in old paint on a bridge, and also in the reflecting film that is sometimes glued to windows. On a gigantic scale it can be seen in Arctic and Antarctic pack-ice when it begins to break up.
A large-scale map of field boundaries can show similar effects. Some of the field boundaries in this picture are curved to avoid very acute angles. The distribution of angles is clearly biassed away from zero and towards ninety degrees. Why do you think this was done?
Here are two more pictures showing field boundaries. In many areas of Britain, stone walls and hedges mark boundaries that may be many centuries old. Some boundaries are almost 2000 years old, where they are the result of the building of Roman roads. Some roads still follow these alignments, often with deviations of no more than a metre or two from the original straight track. The wing looks as though it is based on straight lines as well, but the shadow of the narrow flap on the wide flap reveals a subtle curvature.
Here are more examples of multiple cracks.
If a crack propagates towards another crack at a shallow angle, it will turn towards the earlier crack as it approaches it, as if attracted, because it forms at right angles to the tension. In a sense the cracks reveal a history of the stresses in the glaze. The big frog is one of the series "Raku Creatures" by John Hine. Raku ware was created by Chorjiro Raku, a Korean potter, and his assistants, who emigrated to Japan. The last picture illustrates the history idea, very well, because the helical trend of the cracks around the side of the bowl shows how the bowl was turned on a wheel, rather than moulded.
Unwelcome patterns of cracks are sometimes seen in old layers of paint and other coatings.
The development of cracks like those in glazing can be rather like the development of streets in an old village. Perhaps three or four roads meet, and a ribbon of houses appears along each. Then a road or two may be built between some of these main roads. Over the years a network of roads builds up. In some cases the roads imitate cracks by bending to meet a main road at right angles, to aid visibility for drivers. Older junctions may have quite acute angles, though they are often modified for the reason just given. This type of road growth might happen as in the imaginary example below.
Cracks do not always form at right angles. If there is no hierarchy, there may a tendency to the formation of lines at 120 °, as in soil polygons, the Giants' Causeway, and the interfaces between bubbles. We can also see such effects in the break-up of ice as it melts, on a large scale in the polar regions, or on a small scale on a window, as we see below.
The pictures below show both 120s and 90s.
Nobody would want to make something that broke on purpose, would they? But an electrical fuse is a deliberate weak link in a system, designed to break when fault conditions threaten other components. Fuses come in many kinds, slow-blow, quick-blow, semiconductor, and they all melt when too much current flows. Once fused, they cease to conduct. Other protection devices become almost a short circuit with over-voltage, clamping down the voltage to protect the system.
Mechanical weak links can be useful as well. A glider launched by a wire from a truck or winch on the ground experiences a downward force from the wire, in addition to its weight. This imposes extra bending moment at the wing-roots. To prevent over-stress, a weak link is placed in the cable, near the glider. It protects both cable and glider from damage. By making weak links from a standard material in a standard size, they can be made to perform reliably. In the long run, of course, a weak link may fail from fatigue induced by the cyclic stress. Normally it would be replaced before this happened. Because gliders vary in size and weight, a gliding club may use two or more sizes of weak links, in conjunction with rules about launching speeds. The diagram below shows a typical weak link.
In the Charpy and Izod impact tests, a V-notch is made in a specimen, which is then struck in a controlled way by a heavy pendulum. These tests evaluate the energy needed to fracture the specimen.
Here is a small town in which these processes of growth have been happening over a long period. Sometimes the growth is not organic, as when an area is stripped out and replaced in a modernizing program. Then an entire suburb is often planned at one time. Nevertheless, the roads usually meet at about right angles despite the curves that the designers put in.
When a town or city becomes large enough, it may be furnished with very large roads and ring-roads. These act as huge cracks which cut through between neighbourhoods, making movement between them difficult, except by subways, or by footbridges with long ramps, difficult for people with prams or baggage, old people, and people with disability. Railways have been separating parts of towns for much longer, leading to the expression "wrong side of the tracks", this, in Britain, usually being on the east side, because of the prevailing wind.
The first picture above shows the roads leading west from Cointrin Airport, Geneva, and the second shows Brighton. The A23 to London reaches the left side of the frame, while the A27 runs almost vertically up near the left side. The sandy beach is visible, also the pier, and beyond it the white cliffs leading to Rottingdean and Saltdean. The lines of the railways can also be seen.
The fourth picture above shows Rottingdean, Saltdean and Peacehaven in Sussex, and the white chalk cliffs that separate them from the sea.
The patterns of the fields form a record of history, if only we could read it. When we look at the patterns of the fields in the pictures above, we cannot now discover the alliances, marriages, quarrels or feuds that may have led to the creation of these patterns. We certainly become aware of history when we are on a country road which has too many corners, and seems to go round three sides of a square all too often, and all because some people would not let it through their land.
This view of Paris reminds us that human constructions are sometimes extremely geometrical without using right-angles. La Place Charles de Gaulle, with its twelve radiating streets, is visible in this picture. This deliberately geometrical construction is a complete contrast with the organic and apparently haphazard development of old towns. The route péripherique also makes its mark, but the dominating feature is, of course, the river Seine, its meanders fossilised by concrete walls. The many arrondissements and smaller areas join with dislocations like those between domains in a crystal.
New York city, on the other hand, represents the extreme domination of the rectangle (apart from Broadway and some other streets). This type of construction is found all over the world. It is, of course, not new. Grid layouts, with due care about alignment with the compass and with geographical features, are found in ancient Chinese and Japanese cities. Kyoto's grid is aligned north-south and east-west, in a propitious location, according to the rules of geomancy, with hills to the north, east, and west.
Large numbers of fields or streets are collected together in large units, such as parishes, rural districts, and urban districts. These in turn are grouped into counties. In a large country like the USA, a large number of counties comprises a state. At all levels, from field to nation state, boundaries are sometimes natural, for example rivers or mountain ridges, or artificial, often being straight lines.
During periods of stress or conflict, the effects of boundaries between people can be as divisive and complete as those of cracks in metal. Sometimes their are several boundaries which do not coincide. such as those between ethnic groups, between religious groups, between language groups, and between nation states.
Animals, too, can have quite sharply bounded territories. We might not expect plants to have territories, but among the entries for the Wildlife Photographer of the Year for the year 2000, there is a photograph showing the phenomenon of "crown shyness", which prevents the tops of trees from touching each other, even when they are crowded. Narrow strips of sky are seen between the trees, making a tessellation of brightly outlined polygons overhead. Why do you think this happens?
This image of dragonfly wings shows clearly how the main wing veins turn towards the trailing edge, for the same reason as new cracks turning to join old ones. Even the subsidiary veins tend to touch the trailing edge and the main veins at about 90 degrees, whereas in the tiny cells the angles are often around 60 degrees, minimising the material used, as in a honey-comb. The angle seems to depend on the difference in size, or more accurately, in hierarchical level, between the veins that meet.
No detail is too small for nature to optimise, as long as the overall effect, however tiny, on reproductive probability, is positive. Nature has had plenty of time to get it right. In fact, this design has changed little in 200 million years. The picture below shows an even older design, a damselfly wing. It shows very well the strong leading edge, especially from the root to the nodus, and the more flexible trailing edge, which work together with the thorax to provide the required changes in angle of incidence as the wing flaps. The arrangement of the veins has to satisfy the requirements of expansion of the crumpled soft wings after emergence, and also those of flight. Evolution is affected by every stage of life until reproduction ceases.
This picture shows a mayfly, a member of another ancient order of insects, the ephemeroptera. Mayflies are very feeble fliers, and they fly only for a short time, in order to find a mate. The curvature of the veins towards the trailing edge is very slight.
Another ephemeral flier is the fruit of the sycamore tree. It makes only one flight, from the branch to the ground, and in doing so, it operates in a mode that pilots avoid - autorotation. Autorotation is a stable state in which height is lost at a constant rate, unless power is supplied, as in the autogyro. In a spin, at least a part of the wing area is stalled, and recovery is impossible until true flying is restored. Recovery in an aircraft usually requires settings of the controls which are non-intuitive, and in the early days of flying, many lives were lost before pilots learned the technique. If you pull the wings off a sycamore seed, you will find that it falls much faster than it falls with them on. On a windy day, the slow descent in the spin gives the seeds a better chance to reach a place beyond the influence of the parent tree.
The veins on these wings are like those of the dragonfly - they curve strongly towards the trailing edge, and the leading edge is where the rigidity is greatest. A peculiar aspect of autorotation is that the spinning object can be precisely symmetrical. If you centralize the controls in a spinning glider, you will not necessarily recover from the spin. The direction of the spin may have been decided by some small deflection by the air, though in fact a badly executed turn at low speed is a more likely cause.
Physicists will recognize the spin as an example of broken symmetry, in which a symmetrical system undergoes an asymmetrical development. For example, if you place a small steel ball exactly at the top of a large steel ball, it will eventually roll off in one particular direction.
Here is a large leaf. We can see some of the same effects here as in the dragonfly wing and the pottery. The topology of the veins is especially interesting in the regions where the leaf is about to divide into lobes.
But we must distinguish carefully between the veins in the wings of leaves, and the veins in the wings of insects. Both provide stiffness, but the veins in leaves also provide for the flow of fluids, throughout the life of the leaf, and the angles take this into account. But the flow in the veins of an insect occurs only once, when the wings are expanded, and the emphasis is on the pressure, rather than the flow. And so, in general, if we look at a lot of leaves and a lot of wings, we shall see differences as well as similarities.
In a sense, the stiffening veins in wings and insects are like negative cracks, providing strength where needed. See also Leaves.
If you see in a structure two members that meet at an acute angle, what does that tell you?
The cracks in this piece of wood are governed by the stresses. They resemble the lines of force in the space around two unequal electric or magnetic poles of the same polarity, as in the diagram below. When electric lines of force reach a conductor, they do so at right angles, just like cracks reaching an edge. At the top left of this picture you can see the cracks curving towards the edge like the dragonfly's veins. This piece of wood probably comes from a place near a fork in the trunk. The analogy between the cracks and an electric field must not be taken too seriously: the cracks do not form along the lines of tension - they form at right angles to them. Thus the lines of tension in the wood roughly correspond to the lines of equal potential in the electric field, which are at right angles to the lines of force.
The first picture above, and the diagram next to it, again show a resemblance to an electric field. In a more typical piece of wood the cracks have radial symmetry, apart from the usual small irregularities. In the second example there are a few tangential cracks and some cracks along the direction of cutting. Cracks in dried out ground often form at angles nearer to 60 degrees than 90 degrees, because they can result from simultaneously occurring stresses rather than sequential ones.
Here we see some cracks from the side. Presumably they are radial within the wood, like the ones in the previous pictures. The deviation of the crack near the knot reveals something about the stresses in the wood in that region.
Here is another type of crack in wood, and below are some other shapes. Do some of these pictures remind you of laminar flow of a fluid around an obstacle, or a magnetic field around a superconductor, or even the great red spot on Jupiter?
The diagram below shows a simulation of stress lines around a hole in a large slab, only a part of which is shown. This field of lines also describes laminar flow of a fluid. Evidently the growth of wood around an emerging branch is rather similar. The notches in the lines, caused by the coarse resolution of the original screen, mimic the lines of flow on a larger scale - an example of aliasing.
Cracks are often formed when an object expands differentially, for example, a glass object plunged into hot water, causing the outside to expand faster than the inside. The bark of trees often shows a differential effect, like the crusts of bread and cakes, which of course are subject to tension on a much shorter time-scale. The bark of trees forms cracks because the tree grows faster than the bark. Bark
This section through a tree, left after felling, includes some irregular dark lines. These are not cracks, but the boundaries between regions which have been infiltrated by different fungi. The dark lines are caused by chemicals created by one fungus to repel others.
When things are cracked, they are liable to break if force is applied. When an object is stressed, a crack may zip through it at great speed, so that the break looks instantaneous. After all, a crack is not a physical object, and a high propagation speed can be created by a low separation speed of the material on the two sides. For example, if the opening angle of a crack is 0.1 milliradian, the speed of propagation is 1000 times the speed of movement of the actual material. Given the the kinetic energy of the material is proportional to the square of the speed, we can see that the material does not need to acquire much kinetic energy during propagation.
The picture at left shows a snapped blade of a rusty old pair of scissors.The material is not ductile. The blade was being used as a lever, which was bad practice.
If we look at the shape and texture of surfaces made by different stresses, we see distinct differences. Examples are bending, compression, tension, torsion, and fatigue.
Sometimes an object is made to break deliberately by cutting it partly through and letting it snap off. Here are some tree stumps that were created in this way. The fracture shows where the wood was cut, and also shows the fibrous nature of the wood. Other pictures show natural breaks caused by storms.
Great cracks can open in the earth as a result of faulting and earthquakes. Even quite narrow cracks on hillsides can open up into great valleys as a result of subsequent erosion by water and weather. Cracks can result from sliding (shear) or from pulling apart (tension).
The GreatRiftValley in Africa and the San Andreas Fault are two famous examples of fault action. The Great Glen in Scotland can easily be seen on a map, slicing Scotland in two. The Great Glen is the result of the same type of action as the San Andreas Fault - two pieces of ground sliding past each other. The Great Rift Valley resulted from land pulling apart, allowing a gigantic area to move downwards.
These pictures are of "rift valleys" and a "fault" in a road on an embankment that is slumping. The scale marks are 5 cm apart.
The great tectonic plates are moving at rates up to centimetres per year, in some places sliding past one another, in others pulling apart as hot material wells up from below. The Mid-Atlantic Ridge is a classic example. It even emerges from the ocean, in Iceland, where frequent volcanic eruptions reveal clearly that the earth's activity has not died down. Presumably this activity is fed partly by long lived radioactivity deep below the surface. In other places, plates are moving towards each other, producing great mountain ranges such as the Andes and the Himalayas.
These movements produce the longest cracks on earth. Some are direct discontinuities between plates, where material is upwelling from inside the earth: others are transcurrent faults and transform faults where the location of a spreading ridge undergoes a sideways dislocation. Yet other discontinuities occur where one plate is subducted under another.
Such is the progress of technology that space craft can transmit pictures of cracks on the moons of the giant outer planets.
These cracks were generated in layers of gold leaf on the surface of a vase made of black glass and a cup made of white glass. The effect is rather like the result of the break-up of the seasonal Arctic and Antarctic ice-sheets. The final picture shows the eye of a common frog, Rana temporaria. In this example the gold flecks are more like the Japanese makie technique of depositing gold dust on lacquer. This type of eye pattern is found in frogs of many countries; in some species the pattern looks more obviously like the result of expansion.
Extending this behaviour into the third dimension we see phenomena like the Giants' Causeway and frost polygons.
Glaciers and Crevasses
Click here for some landscapes resulting largely from glacier action following orogenesis.
Near the top of a glacier there is often a crevasse called a bergschrund, where the moving ice and snow have pulled away from the material which remains attached to the rock. Crevasses can be dangerous because you can fall in. If there is a snow bridge the crevasse can be undetectable, even if you are looking for the signs.
But the really dangerous cracks in snow are those that precede an avalanche. Merely walking or skiing across a large slab of snow that overlies a poorly attached layer can start a crack. After that, everything moves with great speed. Escape is unlikely.
Although a large piece of ice can be shattered by a hammer blow, ice can flow slowly under pressure. In a glacier that is hundreds, or even thousands, of metres deep, the pressures are enormous, and the ice flows inexorably downwards, deepening its own valley using rocks embedded in its base.
But where the ice passes a change of direction or slope, if the ice cannot respond quickly enough, crevasses, often huge, open up. These cracks show clearly where the tensile stresses are too much for the ice on the operative time-scale. As a glacier flows over a cliff, pieces of ice as big as a house may break off and fall to the next level. These are bad places for mountaineers. Huge cracks in ice are seen during the annual break-up of the Arctic and Antarctic ice sheets.
In the physics department of the University of Glasgow there is a model of a glacier, consisting of a block of pitch in a wooden box which is divided into two levels. The box has a slot at one side. Since the model was made in the 19th century, some of the pitch has flowed down from the upper level, like a glacier, with a wrinkled surface that looks very like an ice-fall.
These behaviours remind us that the distinction between solids, liquids and gases is not always as clear cut as we sometimes think. At sufficiently high temperatures, the interface between a liquid and its vapour vanishes, and cannot be recreated by any amount of pressure. That temperature is called the critical temperature
Substances like glass will slowly flow if given enough time and stress. They are like supercooled liquids, without the crystalline order than is typical of so many solids. But glass will crack and shatter if stress is too large or too sudden. Lead, too, can creep down a slope. These effects can sometimes be seen in old buildings.
Liquids and gases are not normally associated with cracking, but even they can exhibit discontinuities. We even speak of the crack of a whip. If the position of the sun is right, you can sometimes see the position of an incipient shockwave over the wing of a Boeing 747, as the sharp change in density refracts the light. There is a light band of light along the wing, which wavers as the aircraft repsonds to slight changes in the air. If a supersonic object or a blast wave goes by, the pressure increases almost instantaneously, falls smoothly to a subnormal value, and then returns almost instantaneously to the normal value. Objects that cannot absorb enough energy elastically or plastically will crack or shatter. Water can undergo cavitation behind propeller blades, causing serious erosion as the water collapses back on to the surface.
We see that discontinuities in a material can be caused by static forces which are too strong, or by more dynamic effects on a time-scale which is too fast for the material to respond.
What are the smallest possible cracks? The smallest objects known to take part in collective motion are quarks and gluons, in a quark-gluon plasma. As this is not solid, this medium cannot crack. On the next scale up, nuclear matter is a candidate. There is not enough material in a nucleus to form a solid substance, but in a neutron star it is conceivable that there are solid regions, in which dislocations would be possible. As the neutron star loses energy, there could be internal rearrangements, causing fault lines and star-quakes.
Even larger "boundaries" may exist in the universe, as it appears that most of the matter may be confined to the surface of gigantic "bubbles", with little material inside.
Perhaps space-time is not continuous. There have been suggestions of parallel worlds which are "close" but inaccessible from the dimensions that we experience.
In aerodynamics, narrow gaps can be deliberately introduced in order to increase lift. Many large aircraft have leading edge slats which are moved forward to provide extra lift for take-off and landing. Although leading edge slats (Kruger slats) can be formed by simply hinging parts of the leading edge, providing slots under the slats has a dramatic effect on the air-flow. Some tailplanes have permanent slats under the leading edge. Similar slots are provided between trailing-edge flaps. The wing-tips of large soaring birds have several slots between the feathers.
Other deliberate slots in aircraft are used to make efficient radio aerials without protrusions into the airflow. A slot in a sheet of conducting material can behave like a strip of conducting material in an insulator.
Virtual splits and cracks can appear in mathematics, in catastrophe theory and in chaos theory, where bifurcations suddenly appear as a variable crosses a threshold.
Your train starts to move. Slowly but surely, so very smoothly, it accelerates. You finally get past the train that was next to you in the station, and then your brain does a double-take. It was the other train that was moving, not yours.
We are so used to smooth changes in energy that we take it for granted. Yet this could be an illusion. At the microscopic level, things are very different. Electrons can behave like waves. Waves in a crystal exhibit behaviour that seems strange to us. Certain ranges of wavelength in a periodic cannot be sustained. Because wavelength is related to momentum, and therefore to energy, a crack opens up in the allowed distribution of electron energies in the material.
This gap is not a gap in physical space, but in an abstract space, and it is not a crack in the sense that it can propagate. But it is as real as a crack in a piece of metal. The effects are profound. The distinction between conductors and insulators results directly from it, and the properties of semiconductors depend on it. Most of modern electronics depends on these behaviours. And the understanding of the chemistry and physics of materials has developed largely as a result of quantum theory, which has enabled people to understand atoms and molecules. In these objects, energy is quantised into levels, between which there are states in which the object cannot exist. On the other hand, objects can exist in a superposition of states. Not only are the forbidden energies of atoms and molecules, there are forbidden momenta and energies for the electrons that are free to roam in electrical conductors and semiconductors. The idea that some speeds are not allowed is incomprehensible in classical mechanics. A tunnel diode makes use of this gap.
On the other hand, there is the possibility that particles can pass across regions ins space that are classically forbidden. The Josephson junction use this effect. So does alpha particle decay of atomic nuclei. The classical equivalent would be a boulder in a volcanic crater suddenly appearing at the same height on the outside of the cone, and rolling down the slope.
The very strange quantum theory now affects every aspect of our lives by means of the numerous devices that are based on it. Even large structures like bridges and aircraft have benefitted from the vastly improved knowledge of materials that quantum theory has enabled.
An extreme example of this type of gap was invented by Paul Dirac, who suggested that electrons could have any energy, except for values between mec2 and - mec2, where me is the mass of an electron, and c is the speed of light. The particles corresponding to negative energy were later found to be anti-electrons, now called positrons.
Another type of split appears in the many-worlds interpretation of quantum mechanics, in which both choices of a dichotomy occur, but only on is visible to any one observer. Given the vast number of choices, this idea would apparently lead to an unimaginable number of parallel worlds.
The ultimate boundary is the event horizon between a black hole and the outside world. Everything mentioned in this page can vibrate - even stars can vibrate: can the event horizon vibrate?
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