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Roughness 10x10 micrometer2 AFM image of clay surface (montmorillonite) The photo above is a piece of dried clay which measures 1/1000 millimeters on the side. The photo below is of the Himalayas and measures tens of kilometers. The size ratio between them is 10 billion. There is no confusion to which photo is which. But, take a moment to ponder what is distinguishing them. Well, the upper photo has strange colors, and there is a coordinate system. The lower photo has snow-covered surfaces and clouds. Suppose now that we move the snow and clouds to the upper photo and the strange colors and coordinate system to the lower photo. Are you sure that you would not mistake the clay for Himalayas and vice versa?
Mount Everest What we are trying to convey here is that there is something that is similar in the two photos. What is it? What is it that makes mountains look like mountains - and makes dried clay look like mountains? It is clearly the shape. Now, Mount Everest has a particular shape that allows us to recognize it and say "there is Mount Everest." However, we also recognize mountain ranges that we have never seen before as mountain ranges because they have a shape we expect mountain ranges to have. And it turns out that the dried clay has the same shape. They are all examples of rough surfaces. Is there any way to make these foggy statements precise? That is, clarify precisely what the expression "shape" means in the text above? Yes, there is. However, we cannot escape the language of mathematics to do this. More precisely, we need to use the tools of statistics. We measure the probability that there is a height difference between any two points on the rough surface as a function of the distance between the two points, and then average over all possible placements of the two points. This is the height-height correlation function. Let us call the distance between the two points r and their height difference h. Then the correlation function is p(r,h), and p(r,h)dh is the probability to find a height difference in the range dh about h. The remarkable fact that makes mountains look like mountains and dried clay is the following scaling property: aHp(ar,aHh) = p(r,h) This mathematical statement tells us the following: If we change the distance between the two points by a factor a, we need to change their height difference a factor aH in order to find the same probability as compared to the original distance and height difference. The exponent H is the roughness exponent or the Hurst exponent, and rough surfaces that show this scaling property are called self affine. It turns out that not only are both mountain ranges and dried clay self affine, but they both have a Hurst exponent H which is roughly 0.8. That make their typical shapes very similar. It is surely accidental that the Hurst exponent of these two types of self-affine rough surfaces are so similar. However, brittle fracture surfaces are also self affine (break a piece of chalk and think of the Himalayas) with a Hurst exponent that is also close to 0.8. Now, we have three systems that are widely different, but which has the same roughness exponent. "Einmal ist keinmal, zweimal ist immer." - Once is never, twice is ever. However, we still believe that this numerical similarity is an accident. The Hurst exponent is difficult to measure, and the values for mountain ranges is rather uncertain. This is not so for rough fracture surfaces. The value seems to be quite independent of the material that fractures and it is quite close to the value 0.8 over a wide range of length scales. What is it in the fracture process that makes the fracture
surfaces self affine with a universal roughness exponent H=0.8?
This is still an open question. Is it useful to know that a given surface is self affine? The answer is yes. This knowledge gives us enough information to predict the statistical properties of a surface at one scale given its properties at a different scale. This is for example a situation met when results from laboratory tests are to be applied on full-scale problems. And, self affinity provides strong constraints on any microscopic model of the process that produced the surface. A model for how a self-affine surface is created cannot be right if it does not predict self affinity with the right roughness exponent. We are working on various aspects of rough surfaces. Here are some examples: Roughness
of constrained cracks Wavelet
analysis as a tool to study self affinity Light reflection from rough surfaces |
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