5a Aula Mecanica da Fratura Fractal 1_2
Channel: Lucas Máximo Alves
Date: 2018-09-17

Nesta quinta aula descreve-se um modelo fractal do campo de tensão ao redor de uma trinca.

The use of fractal measure theory to model fracture phenomena is of great interest in the mathematical development of Fractal Fracture Mechanics. However, the fractal models for fracture proposed by several authors presuppose a fractal crack with an infinite range of self-affine scales that cannot be verified experimentally. It is known that a real crack has limits imposed by a finite number of fractal scales ranging from an upper fractal
cut-off scale to a lower fractal cut-off scale. In a real fracture, the finite limits of fractal scales are always compatible with the minimum microcrack and with the length of the macroscopic crack in the sample. Thus, the lower fractal cut-off scale is given by the
characteristic size of a microcrack, while the length of the macroscopic crack on the specimen indicates the upper fractal cut-off scale. Therefore, fractal models of fracture phenomena require a more realistic approach that takes into account the limits of scales and singularity exponents, and hence, the fractal scales that are actually found in a real fracture.
This chapter presents a model of a stress and displacement field around a rough crack tip in brittle and ductile materials, using a pre-fractal. This approach allows for a more realistic
fractal scale of a real fracture. A special stress vector was defined for rough surfaces, and it was found that to consider the pre-fractal also results in a different asymptotic limit for the
singularity of the stress field at the crack tip in brittle materials. The asymptotic limit obtained here differs from the order of singularity presented by other authors. Therefore,
other consequences, such as fracture instability, are also included in the mathematical model presented here. A generalization of the stress fields for cases of brittle and ductile materials
is proposed. Changes in fracture geometry are caused by the roughness at the crack tip, as shown by the mapping of simulated stress fields around a crack tip. The fractal stress field
functions are mapped at different orders of magnitude of singularity, indicating that the qualitative aspects of these fields, alone, do not suffice to determine which model presents
the best fit to experimental results. Therefore, the model is validated based on quantitative experimental measurements.