Geometrical minimum units of fracture patterns in two-dimensional space: Lattice and discrete Walsh functions

We present the geometrical minimum units of fracture patterns in two-dimensional space. For this analysis, a new method is developed from the algebraic approach: the concept of lattice (a type of partially ordered set) is applied to the discrete Walsh functions that have been used to measure symmetropy (an object related to symmetry and entropy) of fracture patterns. We concluded that the minimum units of fracture patterns can be expressed as three kinds of lattice. Our model is applied to the temporal change of the spatial pattern of acoustic-emission events in a rock-fracture experiment. As a result, the symmetropy of lattice decreases with the evolution of fracture process. We find that the pre-nucleation process of fracture corresponds to the subcritical states, and the propagation process to the critical states. Moreover, using a particular mathematical structure called sheaf on a lattice, we suggest the algebraic interpretation of fracture process, and provide justification to regard fracturing as an irreversible process.