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

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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

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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