Evolution of Tiling-like Crack Patterns in Maturing Columnar Joints

Abstract

Fracture or cracking essentially involves the formation of new interfaces. These patterns are usually studied as twodimensional mosaics. The new surface that opens up is in the third dimension, along the thickness of the sample. The thickness is usually very small compared to the lateral dimensions of the pattern. A spectacular and distinctive departure from these everyday examples of cracks are columnar joints. Here, molten volcanic lava, by the sea, cools and cracks under appropriate thermal and elastic conditions, causing the crack system to grow downward, creating long, vertical columns with polygonal cross-section. The focus of this paper is the study of the elongated interfaces of these columns: how the crosssection of their outlines gradually undergoes a metamorphosis from a disordered-looking Gilbert tessellation to a well-ordered hexagonal Voronoi pattern. As the columns grow downward to lengths of several meters (in natural systems), their outline continuously changes, the center may shift, causing the column to twist. For the first time, the evolution of these crack mosaics has been simulated and mapped as a trajectory of a 4-vector tuple in a geometry-topology domain. The trajectory of the columnar joint systems is found to depend on the crack seed distribution and crack orientation. An empirical relationship between the system energy and the crack mosaic shape parameter λ has been proposed on the basis of principles of fracture mechanics. The total system energy shows a power-law dependence on λ with the exponent β ∼ 0.3 and λ ≈ 0.75 at crack maturation. The parameter values are validated by matching the proposed relation with energy estimates existing in the literature. The relation not only matches the visible changes in geometry but also provides a feasible measure of the energy of the system. The geometric energy for the polygonal mosaics in the transverse section has also been estimated as a function of time. The geometric energy moves toward a minimum as the mosaic becomes more Voronoi-like at maturation.

Three-Dimensional Modeling of Polygonal Ridges in Salt Playas

“…18 The crack pattern and therefore the salt ridge pattern becomes Voronoi-like. 19 The shifting salt ridges are taken as a transient that stabilizes to a steady state once the crack mosaic reaches maturation. Thus, while in the transient state, the evaporation and consequent crystallization leaves a deposit of surface salt layer; the salt ridges grow visibly only once the underlying crack mosaic reaches a steady matured state.…”

“…The Voronoiness of a real mosaic can be measured by comparing it with the Voronoi diagram generated using the centroids of the polygons as seeds. 19 The ■ SIMULATION MODEL Initial Cracks. As water evaporates from clay or soil surfaces during desiccation, the matrix (clay/soil) develops stress due to capillary pressure in the pores.…”

Evolution of Polygonal Crack Patterns in Mud When Subjected to Repeated Wetting-Drying Cycles

“…Therefore it is convenient to express the energy in terms of the shape parameter, i.e., the iso-perimetric ratio of the tessellation, λ. The energy can be expressed in terms of λ as follows [16] -…”

“…It measures the distance between two subsets of a metric space. We can compare a real crack with the corresponding Voronoi diagram generated by using the centroids of the crack peds as the seeds [16]. 1 A related construction, the Delaunay Triangulation, is a triangulation of a finite set of points F with the Empty Circumcircle Property (no point in F is contained in the interior of any triangle of the triangulation).…”

“…In the rough Voronoi tessellation of mud cracks in summer, the seeds of the Voronoi regions are expected to be the local stress points due to desiccation; while in the giant columnar joints formed during the cooling of basaltic lava, the temperature of the individual column is highest at the seeds of the Voronoi regions that are prominent in their transverse cross-section. It has been reported that columnar joints form so as to release maximum thermal stress in order to minimize the system energy [15,16]. The abundance of Voronoi tessellation in efficient network systems in different spheres of life, prompts a closer inspection of the fundamental driving forces responsible for such a geometry.…”