Four Sided Domains in Hierarchical Space Dividing Patterns

Abstract

The cracks observed in the glaze of ceramics form networks, which divide the 2D plane into domains. It is shown that, on the average, the number of sides of these domains is four. This contrasts with the usual 2D space divisions observed in Voronoi tessellation or 2D soap froths. In the latter networks, the number of sides of a domain coincides with the number of its neighbors, which, according to Euler's theorem, has to be six on average. The four sided property observed in cracks is the result of a formation process which can be understood as the successive divisions of domains with no later reorganization. It is generic for all networks having such hierarchical construction rules. We introduce a "geometrical charge," analogous to Euler's topological charge, as the difference from four of the number of sides of a domain. It is preserved during the pattern formation of the crack pattern.

Self-Replicating Cracks: A Collaborative Fracture Mode in Thin Films

“…In the case of a stiff substrate this interaction distance is on the order of the thickness of the coating. Such familiar hierarchical disordered patterns are for instance observed in dry mud [8,9], in the glaze of ceramics or even in plant venation and urban networks [10]. Delamination may eventually occur after the previous fracture pattern has been established [11].…”

Repulsion and Attraction between a Pair of Cracks in a Plastic Sheet

“…The stress field around the crack tip is then modified by the presence of the other cracks leading to kinked trajectories. In hierarchical fracturing processes [13], it is usually observed that cracks coalesce around a 90°angle, as for example in dessication [14][15][16] or in fault dynamics [17][18][19][20]. It has also been shown that two collinear approaching cracks (submitted to a uniaxial stress in mode I) can repel each other instead of merging tip to tip [21], resulting in a curved, "hookshaped" path.…”

Sculpting of Rocks by Reactive Fluids

“…Part c shows the statistical analysis of the fracture pattern. The number of sides for each domain of the fracture pattern shows clear peaks for the four sided domains, which is characteristic of hierarchical fracture patterns (Bohn et al, 2005). Triple junctions were found to comprise as much as 89.0% of the total junctions in the network.…”

“…In such patterns fracture angles close to 90 ° and 180 ° (i.e. T-junctions) and four sided domains dominate (Bohn et al, 2005). The size distribution of unfractured domains also follows a log normal distribution (Iyer et al, 2008;Plümper et al, 2012).…”

“…The evidence includes a fracture pattern that strongly resembles the hierarchical patterns that form during slowly driven volume changes in brittle materials, characterised by a high frequency of T junctions, 4 sided domains and log normal fracture length distribution. Bohn et al (2005). Long fractures, whose origins are difficult to determine, are labelled as Generation 1.…”