FRACTAL CHARACTERIZATION OF 3D SURFACE-ROUGHNESS

This paper reinvestigates some ideas concerning the fractal-based characterization of surface roughness. In a first stage we recall the relations between notions connected with the non differentiability of certain functions: Lipschitz-Holder exponent, fractal dimension and spectral exponent. We show that the existence of a Lipschitz-Holder exponent for each point of the surface is a sufficient condition to have a well-defined fractal dimension. This property does not imply that the underlying random function is non stationary, or some global self affinity property. However, around each point x, the existence of a Lipschitz-Holder exponent H implies that locally the function z* (DELTA) = z(x + DELTA) - z(x) is a statistical self affine function, that is to say a function for which var (z*(lambda DELTA)) = lambda 2H var (z*(DELTA)) in a disk, the radius of which is characteristic of the surface. All these conclusions are supported by a computation based upon the use of the Weierstrass-Mandelbrot function and the Weierstrass one. Some 3D simulations of fractal surfaces are also presented on the basis of an original 2D extension of the Weierstrass function.