Fractal objects derive from many interface phenomena, as they arise in, for example, materials science, chemistry and geology. Hence the problem of estimating fractal dimension becomes of both theoretical and practical importance. Existing algorithms implement the standard definitions of fractal dimension directly, but, as we show, often give unreliable results when applied to digitized and quantized data. We present a new algorithm for estimating the fractal dimension of surfaces - the variation method - that is more reliable and robust than the standard ones. It is based on a new definition of fractal dimension particularly suited for graphs of functions. The variation method is validated with both fractional brownian surfaces and Takagi surfaces, two classes of mathematical objects with known fractal dimension, and is shown to give more accurate results than the classical algorithms. Finally, our new algorithm is applied to data from sand-blasted metal surfaces.

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Quantitative fractography is often used to study material failure mechanisms. During calculation of surface or profile roughness parameters, the magnification used in obtaining fractographic data is found to influence the value of the parameters. Fractal geometry has been developed into a tool capable of defining surface and profile topography without sensitivity to magnification, and several studies have related fractal dimension (OF) to other physical or mechanical properties. In this study, we obtained the fractal dimension of profiled fracture surfaces of one glass and three proprietary dental porcelains. The fracture toughness ( K J of these materials was also measured using the indentation-strength method. Results show the surfaces to be fractal. No quantitative relationship between fractal dimension and toughness was found. Differences in K,, were demonstrated between some materials. It is postulated that the size range within which fractal dimension can be defined as constant is dependent on the toughening mechanism, and that the relationship between K,, and D, cannot be identical for all materials.