Self-similarity in concrete fracture: size-scale effects and transition between different collapse mechanisms

The paper revisits the fundamentals of fractal modeling of size-scale effects evidencing hypotheses, strengths and the weaknesses of the approach. The authors give a fractal explanation of the size-scale effects by using the renormalization group theory, and present a fractal cohesive crack model. Among various demonstrations proposed to explain the slopes of multifractal scaling laws, the fractal and statistical one are both discussed. It is shown that the material parameters determining the cohesive law are size-dependent. However, the size effects can be explained assuming that the damage occurs within a fractal domain. The scaling laws are obtained for ultimate strength, critical strain and fracture energy. Then two multifractal scaling laws are limited to type 1 size effect, i.e., only to structures of positive geometry (unnotched), failing at fracture initiation, and not to type 2 which occurs in unnotched or notched structures that reach the peak load only after extended and stable crack growths. In the case of type 2 size effect, several other models are revisited which predict the scaling law. These models are often based on a similarity approach, since similar crack patterns are assumed in beams of different sizes. This point is analyzed for the scaling in failure of reinforced concrete (RC) beams. The authors discuss the interactions between three collapse mechanisms occurring in the RC beams failing in shear, i.e., flexural, diagonal tension and crushing ones. It is shown that the diagonal shear failure is the intermediate (transitional) failure mode between two other failure types, i.e., flexural fracturing with yielding of the reinforcement and concrete crushing.