Morphological Fractal Dimension Versus Power-law Exponent in the Scaling of Damaged Media

Fractal Geometry has been widely used for the description of irregular phenomena in various scientific fields recently. In the subjects concerning fracture system characterization, fractals represent the fracture surfaces in two or 3D problems. In the last few years the fractal geometry of crack networks in damaged materials has been statistically characterized by two power laws, respectively, describing the spatial distribution of crack barycenters, and the crack length distribution. In this article, we explore the potential of the latter power-law. Merely using such statistical model to describe the population of cracks, besides providing a theoretical basis for explaining lower limits to the b-value both in seismicity and in acoustic emission (AE) tests, we find a simple relation between b and the fractal dimension D of the crack network. As a result, the b-value analysis in AE monitoring tests permits evaluation of the dimension D of the damaged domain. This method of evaluating D is herein applied to a concrete specimen in compression, subjected to AE monitoring, loaded up to failure. In this test, the characterization of the fracture process through analysis of AE signals emerging from the growing cracks has been performed in a post-processing environment, using two different procedures. In fact, besides the two-point correlation algorithm introduced by Grassberger and Procaccia, the damage process has been evaluated through the b-value analysis. Both procedures make it possible to evaluate the dimension D of the damaged domain, i.e., the fractal dimension of the crack network. The obtained results are consistent with our understanding of damage phenomenon.