Estimating theoretical strength of brittle materials using fractal geometry

The estimation of the theoretical strength of materials is important in order to assess the potential for improvements in processing and design of structures, especially with the increased interests in nanostructured materials. Most estimates are based on knowledge of inexact potential energy functions. Fractal geometry presents a mathematical approach to the understanding of fracture in materials at all length scales. The fractal dimension, which is a quantitative characterization of the tortuosity of the fracture surface, is directly related to the toughness of a material. Fractal geometry provides a framework by which bond rupture can be described as a series of bond reconfigurations. This article shows how fractal geometry can use this framework to estimate the theoretical strength of materials based on crack tip geometry and generated fracture surface without any assumption of the shape or type of potential energy function. The estimates are between the commonly quoted values of E / π and E / 8.