Cracking of heterogeneous brittle materials like portland cement concrete (PCC) is usually caused by overstressing under tension. Unlike laminated and periodic materials such as fiber reinforced composites and polymers, cracking in heterogeneous brittle materials takes the form of irregular geometries. The objective of this paper is to investigate the quantitative features of multi-scale two-dimensional Julia sets for their use in simulating artificial cracks in heterogeneous brittle materials. Three parameters in the creation of Julia sets are identified in this study, including the maximum iteration number n, the real part of Julia coefficient ( \(c'\) ), and the imaginary part of Julia coefficient ( \(c''\) ). Fractal dimension of each Julia set is measured by the box counting method. Crack area A of each Julia set is used to quantify the severity of an artificial crack for engineering applications, as well as crack length L. Threshold value \(\mu \) and the radius ratio of crack growth r are used to control crack opening width, crack area, and crack length. With an assumed descending relation between threshold value and the radius ratio of crack growth, a normalized simulation time t (depending on threshold value) is proposed to generate fractal cracks at different stages. From our parametric studies, it is found that crack area is exponentially related to fractal dimension by a power factor of 12.15, while fractal dimension is linearly related to the maximum iteration number. Both the increase of \(c'\) and \(c''\) generally leads to the linear decrease of fractal dimension but a linear increase of crack area. Crack area is affected by \(c'\) , \(c''\) , the maximum iteration number, threshold value and the radius ratio of crack growth. Crack opening width w is affected by iteration number and threshold value. Crack length L is affected by threshold value and the radius ratio of crack growth. A six-step design procedure is proposed for systematically generating fractal cracks at different stages of cracking.
