An Introduction to Fractal Analysis

In failure analysis, fractographs provide both 2D and 3D surface renderings that enable one to qualitatively characterize the type of failure. Some qualitatively described surface features such as mirror-mist-hackle in certain materials can also be used to correlate these features with material properties, such as the stress of a material at the point of failure. Another measure of surface ruggedness that has been used is a fractal dimension. The fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (fractal pattern) changes with the scale at which it is measured. Thus a fractal dimension is a quantitative measure of surface ruggedness. One can allocate a fractional dimension to the topological dimension of 1, 2, or 3. Morphological features, the ruggedness of which can be described by fractal dimension, are also called 'Fractals'. Determination of a fractal dimension requires a measuring tool to generate a 1D outline, 2D surface, or 3D representation. Most analyses are restricted to 1D and 2D. Scanning Electron Microscopy (SEM), Atomic Force Microscopy (AFM), Light Microscopy (including line section) and Scanning White Light Interferometry (SWLI) are all candidates for measuring surface profiles. A number of materials have been investigated such as brittle (silica glass, rock), ductile (aluminum alloy, steel, SS) and quasi-brittle (mortar, wood, composites) . These have been found to have certain self-affine characteristics. The fractal dimensions (f_D) and their characteristics help quantitatively ascribed properties and features to material composition. Some of these are stress at failure, fracture toughness, orientation/direction of applied stress, wear/erosion/corrosion type, roughness when other methods fail, temperature difference of polymer fracture surfaces, correlation with elemental distribution, heat treatment effects on surfaces, and quantification of intergranular v. intragranular fracture.