The Self-Made Tapestry Phần 6 pot

This is a very simplistic picture of fracture: for one thing, it insists that one bond must always break at each point along the crack with each time stepbut in reality there is no reason why this has to be so if the stress isn't large enough. But all the same, the model provides some indication of why cracks might have a fractal branching structure. A better model would make allowance for the fact that bonds can stretch a little without breaking: they are not like rigid rods, but more like springs. This means that, each time a bond breaks, it will release stress in the immediate vicinity and the surrounding bonds can relax somewhat. Fracture models that modify the dielectric breakdown picture to allow for bond stretching and relaxation have been developed by Paul Meakin, Len Sander and others, and they can generate a range of different fracture patterns depending on the assumptions made about bond elasticity and so forth; an example is shown in Fig. 6.13. This crack has a much less dense network of branches than those generated by the 'pure' dielectric-breakdown model, and to my eye it looks much more like the kind of pattern you might finds creeping ominously across the ceiling. The fractal dimension is 1.16, showing that the crack is less like a two-dimensional cluster and more like a two-dimensional cluster and more like a wiggly line. Fig. 6.13 Crack formation can be modelled by a modified form of the dielectric breakdown model that allows bonds to stretch and relax. This can generate more tenuous, almost one-dimensional branching patterns. (Image: Paul Meakin, University of Oslo.) Patterns in the dry season In all of these examples the crack starts at a single point and spreads from there as the material is stressed. But not all cracks are like that. Think of the fragmented hard mud of a dried-up pond during a drought (Fig. 6.14). What has happened here is that, as the wet mud at the pond bottom has become exposed and dried, the tiny particles have all drawn closer together and aggregated into a compact layer. In effect, the wet mud has been exposed to an internal stress that acts at all points as the material contracts. This means that cracks have been initiated at random throughout the system and have propagated to carve up the mud into islands. This kind of cracking due to uniform shrinkage (or expansion) of a thin layer of material is a common problem in engineering. It might happen to a layer of paint as the material on which it sits expands or contracts because of temperature changes. Surface coatings Page 150 are commonly deposited in a 'wet' form onto an engineering component to protect it or to modify its surface properties (to make it more wear-resistant or less reflective, for instance), and these coatings then shrink as they dry, while the underlying surface retains the same area. Integrated microelectronic devices often incorporate a thin film of one material (an insulator perhaps) laid down on top of another (a semiconductor, say) in which the spacing between atoms is slightly differentso to maintain atom-to- atom bonding at the interface, the overlayer has to be slightly expanded or compressed, and the film is uniformly stressed and liable to crack. Thus there are many very practical reasons for wanting to understand the fracture patterns produced in thin layers of material that are uniformly stressed by expansion or shrinkage. Fig. 6.14 When a thin layer of material is stressed as it shrinks, it can fragment into a series of islands of many different size scales. Here this process has occurred in drying mud. (Photo: Stephen Morris, University of Toronto.) Arne Skjeltorp from the Institute for Energy Technology in Norway has explored a model experimental system for this type of fracture, consisting of a single layer of microscopic, equal-sized spheres of polystyrene, just a few thousandths of a millimetre in diameter, confined between two sheets of glass. This is an excellent model for the shrinkage of dried mud in a pond bed, because the interactions between the particles are directly analogous to those between silt particles, and because the layer of microspheres, deposited from a suspension in water, likewise contracts and cracks as the water evaporates. Skjeltorp found that these layers of spheres fracture into complex 'crazy paving' patterns, highly reminiscent of dried-up river or lake beds, as drying progresses. Figure 6.15a shows the early stages of the process, and Fig. 6.15b and c show the final pattern at two different scales of magnification. The first thing to notice is that the cracks have preferred directions, at angles of 120° to one another (this is particularly evident in Fig. 6.15a). This reflects the symmetry of the underlying lattice of particles, in which they are packed in a hexagonal array. The cracks tend to propagate along the lines between rows of particles, as can be seen clearly in c. The particles in mud are likely to be packed together in a much more disorderly fashion, and so the shapes of the final islands are less regular (Fig. 6.14). The second thing to note is that the pattern looks similar at different scales of magnification (this can be seen to some degree by comparing Fig. 6.15b and c, except that in the latter we lose the smallest scales because we are reaching scales comparable to the size of the particles themselves). This property is, as we now know, a characteristic of fractal patterns. And indeed these fracture patterns are fractal over the appropriate range of scalesSkjeltorp found that they have a fractal dimension of about 1.68, slightly lower than that of DLA clusters. Can we reproduce these patterns using the sort of simple probabilistic models of fracture described above? We can indeed. Paul Meakin has adapted the 'elastic' dielectric breakdown model so that it is an appropriate description of Skjeltorp's thin layers of polymer microspheres uniformly stressed by shrinkage. It was important in this model to include the fact that the microspheres are attracted weakly to the confining glass platesthis, Skjeltorp points out, means that the cracks propagate further than they would do otherwise because a crack shifts the spheres away from their initial point of binding to the glass and so sets up additional stresses that drive the crack onward. Allowing for this effect, Meakin found that the model produces crack patterns similar to those observed in the experiments (Fig. 6.16). What should we conclude from all of this about the web-like branches of cracks? The detailed investigations of the stresses around a rapidly propagating crack tip performed in recent years have enabled us to understand why it is that these fast cracks tend to split into branches: there is a dynamical instability which makes simple forward movement of the tip untenable. Beyond this threshold there is an underlying unpredictability in the motion of the crack tip, so that the crack carves out a jagged path that splits the material into rugged (and Page 151 Fig. 6.15 The cracks in a layer of microscopic polymer particles as the layer dries. Because the particles are packed in a hexagonal array, the cracks tend to follow the lines between rows of particles and so diverge at angles close to 120°. This is particularly evident in the early stages of cracking (a). The final crack pattern (b, c) looks similar at different scales, until we reach a scale at which the discrete nature of the particles makes itself evident (c). The region in frame b is about one millimetre across; that in c is ten times smaller. (Images: Arne Skjeltorp, Institute for Energy Technology, Kjeller.) generally fractal) fracture surfaces. Randomness and disorder in a material's structure provide a background 'noise' that can accentuate the pattern. While in some ways fracture remains a unique and immensely challenging (not to mention practically important) problem, it is nonetheless possible to develop models that seem capable of describing at least some kinds of breakdown process while establishing a connection to other types of branching pattern formation. A river runs through it When biologist Richard Dawkins, in his book River Out of Eden, compared evolution to a river, his metaphor was based on pattern. Like a river, evolution has its luxuriant branches (Fig. 6.17), a host of tributaries arrayed through time and converging to the broad primary channels of life in the distant past. (Don't look at the analogy too closely, however. It has its strong Page 152 points, but a river branches upstream, whereas if time is evolution's directional arrow then its bifurcations are distinctly downstream. And some biologists, like Stephen Jay Gould, have spent their lives arguing vigorously that evolution has no 'direction' at all.) Fig. 6.16 A modified form of the dielectric breakdown model is able to reproduce the fracture patterns seen in contracting thin films. (Image:Paul Meakin.) Fig. 6.17 The phylogenetic trees that trace out evolutionary relationships have something of the branching structure of a river delta. Older phylogenies, such as that shown here due to Ernst Haeckel, tended to over-emphasize this pattern, however; Stephen Jay Gould cautions against regarding evolution as a force of increasing diversification. The curious thing about a river network is that it generally grows in the opposite direction to the way the water flowsfrom the tips of the tributaries into the surrounding rock. There is a very real sense in which we can regard it as a crack, propagating slowly (quasistatically) through the rock of a hill or mountain range. Yet the physics of this growth process are at face value very different from those of a crack spreading through stone. Streams grow back from their tips as water from the surrounding slopes flows down into the channel, wearing the rock away little by little. All the same, the result (Fig. 6.18) is a pattern that looks strikingly like a crack, or for that matter like a fractal aggregate or an electrical dischargebut on scales perhaps a million times greater. Already we can smell universality afoot. To what extent is it really so? Fig. 6.18 River networksgeomorphological cracks on a grand scale? (Photo: Jim Kirchner, University of California at Berkeley.) For geomorphologiststhose who study the shapes of landscapesmany decades ago, there was none of the modern language for describing or conceptualizing branched patterns like this, and they struggled to invent one. The first attempt to do so was made by the Page 153 American engineer Robert E. Horton in the 1930s. He formulated a series of 'laws of drainage network composition' which were held to be universal for stream networks. Horton's scheme was modified by A. N. Strahler in 1952, who classified the elements of a network by assigning them an 'order' that signifies their position in the hierarchy of branches. The outermost streams, which themselves have no tributaries, are first-order. Where two first-order streams join, the resulting stream is second-order; and in general, the meeting of two streams of a given order signals the beginning of a stream of next-highest order (Fig. 6.19). If a lower-order stream flows into a higher-order stream, the former terminates but the latter's order is unchanged. Fig. 6.19 The hierarchy of river network elements in Strahler's modification of Horton's classification scheme. Each branch is assigned an order that increases downstream. This sensible but somewhat arbitrary classification scheme enabled Horton to identify some general rules governing stream networks. His 'law of stream numbers' states that the number of streams of a particular order decreases with orderthere are fewer higher-order streams than lower-order. You could probably guess this rule from Fig. 6.19, but Horton was able to express it with mathematical precision: the number of streams of order n is roughly proportional to the inverse of a constant raised to the power n. In other words, this law of Horton's is a scaling law. Another way of expressing this relationship is to say that the number of streams in each order is a constant times the number in the next-highest order. The number of first-order streams in a particular network might, for example, be four times the number of second-order streams, which is itself four times the number of third-order, and so on. [...]... Richardson found that the apparent length of these boundaries depended on the scale of the map that one used to make the measurement: small-scale maps show more detail than large-scale ones, and so capture more of the nooks and crannies, making the total length seem longer If the logarithm of the length of the boundary is plotted against the logarithm of the length of the yardstick, the points fall on... percolation, however, the pore network of the surrounding medium imposes its own pattern, and the invading fluid advances through this network in a densely interweaving pattern (Fig 6. 20) The probability of the invading fluid displacing the other is dependent on the size of the pore through which the fluid passes, since this modifies the pressure at the displacement front If the pore network is highly disordered,... islandscross-sections of the peaks separated by gaps (b) So it's not quite so straightforward to measure the fractal dimension of a self-affine surface One way is to look at many cross-sections like Fig 6. 28a, by taking cuts through the surface, and to see how their length depends on the length of the ruler (see Fig 6. 26) The fractal dimension of the wiggly crosssections can then be related to that of the surface... careful look at the cross-sectional profiles of the model ridges reveals a deeper similarity with mountain ridges than is immediately apparent The rather flat ridge shown in Fig 6. 31 a doesn't obviously resemble the jagged section of the Dolomites in Fig 6. 31 buntil you exaggerate the vertical scale of the ridge's profile, whereupon the two look remarkably alike (Fig 6. 31 c) You might ask whether it's really... solve it, and in the end we are forced to go back, like the French mathematician Jean Leray in the early twentieth century, and gaze instead at the real thing: the eddies of the Seine as it flows beneath the Pont Neuf in Paris There are patterns in there, to be surewe observe the swirling vortices being born and swallowed upbut how can one formulate an exact description of them? Many of the greatest scientists... analogous to the stress imposed on a fracturing material or the electrical power fed into a spark discharge, is the kinetic energy of the rainwater flowing down the contours of the landscape This energy input to the system is greatest where the water flows fastest and most abundantlythat is, where steep slopes converge They do so at the head of the stream channels, where water flowing across the rock surface... way of getting at the fractal dimension of an object, which is an invariant geometrical property of the way it occupies space Fig 6. 27 Lewis Fry Richardson found that the lengths of many coastlines and borders depend on the size of the measuring stick, increasing as the stick gets smaller When the logarithm of the apparent length is plotted against the logarithm of the stick length, the measurements... have much the same kind of fractal structure as cracks and DLA clusters Invasion of the highlands Recall that in both the latter cases, growth of the pattern from the branch tips is more probable than from deeper within the 'tree' For cracks this is because the stress is greatest at the tips, just as, within the dielectric breakdown model, the electric field around the discharge tips is largest The energetic... edges, their area would have increased in proportion to the square of their perimeter, and a graph of the logarithm of the area against the logarithm of the perimeter would be a straight line with a slope of 2 Because they (like the surface itself) were rough, fractal objects, however, their areas increased more rapidly with increasing perimeter, and the log-log plot had a slope of 2.28which is the fractal... at each step The next bond to break was always chosen to be the weakest one along the perimeter of the cluster You can now see that this model describes essentially the same process as the dielectric breakdown model, except that the next bond to break is always, rather than most probably, the weakest It is simply another slight variant on the model of fracture in a disordered solid Fig 6. 20 Invasion . pattern (Fig. 6. 20). The probability of the invading fluid displacing the other is dependent on the size of the pore through which the fluid passes, since this modifies the pressure at the displacement. diversification. The curious thing about a river network is that it generally grows in the opposite direction to the way the water flowsfrom the tips of the tributaries into the surrounding rock. There. progresses. Figure 6. 15a shows the early stages of the process, and Fig. 6. 15b and c show the final pattern at two different scales of magnification. The first thing to notice is that the cracks have