Roughness exponent

A. Hansen, J. Origin of the universal roughness exponent of brittle fracture surfaces stress-weighted percolation in the damage zone. Phys. Rev. Lett. 90, 045504 (2003)... 

Fig. 3.6. A typical computer-generated crack surface, with the roughness exponent ( = 0.85 (from Roux 1994).

Fig. 3.19. Molecular dynamic simulation results for the fracture propagation in amorphous structures (with Lennard-Jones potential) show that the average fracture velocity crosses over to a higher value (ufinai from Uinitiaij indicated by the dotted lines) at the late stages of growth, as the crack size exceeds the typical size (correlation length) of the voids in the network. The inset shows that a corresponding crossover in the fractured surface roughness exponent also occurs along with the crossover in the fracture velocity (from Nakano et al 1995).

Assuming constant normal pressure in the contacts, which is equivalent to a linearity between L and A, the static friction coefficient obeys the following general scaling laws for rigid objects where roughness exponent H = 0 applies ... 

Here, the mean height (/z)=0, Eq. (6.1) is calculated on an image with scan length L, f is the spatial frequency (i.e. the inverse distance), and /max is the Nyquist frequency. The analysis for the relevant scaling properties (viz. correlation length, roughness exponent, etc.) can be carried out either on the roughness (6-7)... 

Neufeld et al. (1999) have shown that the roughness exponent a of the decaying chemical field is a function of the decay rate and of the Lyapunov exponent of the advection. In the large Peclet number limit we can neglect diffusion and set D = 0 so that the concentration field can be described by the Lagrangian representation (6.13) that follows the chemical dynamics within the fluid parcels advected on chaotic trajectories in the flow... 

The roughness versus scanned area pattern is characteristic of a given material and defines a fractal dimension, dp., which is evaluated as dp.= 3—a where a is the so-called roughness exponent that can be calculated as the slope of roughness versus scan size in a double log plot [42]. 

Let us consider briefly the implications of this model. We discuss the two-dimensional case d = 2), which is closer to physical situations since it corresponds to a two-dimensional surface of a three-dimensional solid. In this case, the roughness exponent a is zero, meaning that the width of the surface profile does not grow as a power of the system size this does not mean that the width does not increase, only that it increases slower than any power of the linear size of the system. The only possibility then is that the width increases logarithmically with system size, ic In L. This, however, is a very weak divergence of the surface profile width, implying that the surface is overall quite flat. The flatness of the surface in this model is a result of the surface tension term, which has the effect of smoothing out the surface, as we discussed earlier. 

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