A new mathematical tool for analyzing the fracturing process in rock: Partial symmetropy of microfracturing

We have shown that the symmetry of fracturing process in a macro-scale can be quantified by using the concept of symmetropy (an entropy-like measure of symmetry). Here we extend this approach to examining the symmetries in a range from small (partial) scales to larger (whole) scales: this approach is called partial symmetropy (PS). To check its applicability, we consider one illustrative example, the temporal change of the spatial patterns of acoustic-emission events in a well-documented rock fracture experiment. Our results are summarized as follows: (i) the PS enables us to distinguish the nucleation phases from the other phases such as the pre-nucleation and the propagation phases; (ii) the variation of the PS shows that the fracturing process is associated with a type of phase transitions from the subcritical state to the critical state; (iii) the scale dependence of the PS reveals the presence of the sandwich structure that consists of order and non-order in the evolution of the fracturing pattern; (iv) within the framework of non-extensive Tsallis entropy we develop the PS concept, and show that the degree of non-extensivity on a large scale increases immediately after the nucleation. The results shown in the present paper are not obtained by taking the traditional fractal approach, nor using only a simply symmetropy. We therefore propose that the PS is a useful tool to providing a strategy for describing qualitatively the phase changes in the observed fracturing process.