Fractal analysis of the three-dimensional fracture surfaces in materials by the box-counting method

摘要

Fractal geometry created by Mandelbrot [1] enabled the quantitative description of microstructures and fracture surfaces in materials by the fractal dimension. Almqvist [2] proposed the variance method for the estimation of the fractal dimension, D_v (2 ≦ D_v ≦ 3), of the three dimensional surface on the images (larger than about 300 * 300 in pixel) obtained by scanning probe microscopes (SPM). He estimated the fractal dimension (D_v) from a relationship between the average variance, {σ(B)}~2, and the size of the measurement boxes, B, in a given image ({σ(B)}~2 ∝ B~(2(3-D_(v)))). However, the fractal dimension of the three-dimensional surface may depend not only on the image size but also on the algorithms for the fractal analysis [1,3], and may be correlated with surface microstructures and properties of materials [4-6]. Further, the SPM are not suitable for the characterization of very rugged surfaces with many bumps or ridges.