Combinatorial topology and geometry of fracture networks

Roy, Aniruddha; Haque, R. A. I.; Mitra, Atish; Tarafdar, Sujata; Dutta, Tapati

doi:10.1103/physreve.105.034801

Cited by 7 publications

(12 citation statements)

References 71 publications

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“…The 2-dimensional tessellation of the top surface has been done following the (a) Gilbert algorithm and its variants and (b) the Voronoi algorithm. 1 The pattern of the disordered crack mesh has been varied by changing crack seed distribution, crack seed density, and crack orientation. The time development of the polygonal crack mesh mimics a map of successive transverse layers of a columnar joint system as it advances vertically downward from the surface layer.…”

Section: ■ Materials and Methodsmentioning

confidence: 99%

“…These patterns are usually studied as twodimensional mosaics. 1 The new surface, which 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.…”

Section: ■ Introductionmentioning

confidence: 99%

“…5 T h i s c o n t e n t i s To distinguish between crack mosaics of different materials, in an earlier work the authors have proposed a topologygeometry combinatoric (n, v, D, λ) that is characteristic to a crack mosaic. 1 If a node of the crack mosaic is the vertex of n polygonal cells or crack "peds", the node is said to have degree n and is termed "regular", else it is termed "irregular". Each one-dimensional face of the mosaic is called an "edge".…”

Section: ■ Introductionmentioning

confidence: 99%

“…On analysis of about 22 real crack mosaics of different materials, and several variations of simulated crack mosaics, the authors had noted that topologically, crack mosaics belonged to the category of either Gilbert or Voronoi cracks. 1 Inspecting the mathematically allowed space of (n, v, D, λ), it is observed that real crack meshes are mostly clustered around the Gilbert crack representative point with only a few representative points of crack mosaics clustered around the Voronoi crack descriptor; much of the allowed domain space remains empty of real crack mosaic points. Though columnar joints start as Gilbert cracks and ultimately mature to Voronoi cracks, the trajectory of their evolution in the (n, v, D, λ) space remains to be mapped.…”

Section: ■ Introductionmentioning

confidence: 99%

“…Desiccation crack patterns in drying films or slurries is a common phenomenon. These patterns are usually studied as two-dimensional mosaics . The new surface, which opens up, is in the third dimension, along the thickness of the sample.…”

Section: Introductionmentioning

confidence: 99%

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Evolution of Tiling-like Crack Patterns in Maturing Columnar Joints

et al. 2022

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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.

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Section: ■ Materials and Methodsmentioning

confidence: 99%

Section: ■ Introductionmentioning

confidence: 99%

Section: ■ Introductionmentioning

confidence: 99%

Section: ■ Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Evolution of Tiling-like Crack Patterns in Maturing Columnar Joints

et al. 2022

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Evolution of polygonal crack patterns in mud when subjected to repeated wetting–drying cycles

Haque

Mitra

Tarafdar

et al. 2023

Chaos, Solitons & Fractals

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Three-Dimensional Modeling of Polygonal Ridges in Salt Playas

Ibne Haque,

Mitra,

Dutta

2024

Langmuir

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In this work, we investigate the formation of the curious polygonal salt ridges that tessellate salt playas worldwide using suitable three-dimensional modeling and simulation of the dynamical processes that are responsible for their formation. We employ the principles of fracture mechanics under cyclic wetting and drying, fluid and mass transport in fracture channels, and processes of crystallization and self-organization to finally replicate the almost Voronoidal pattern of salt ridge mosaics observed in playas. The model is generic and applicable to playas having different salt compositions, as the effect of the salt diffusion coefficient and critical salinity for crystallization are factored in. The final pattern of polygonal salt ridges obtained by simulation visually resembles the geometry of the salt ridges reported in the literature. A single equation describing the time of first crystallization of salt in terms of evaporation suction pressure P, diffusion coefficient D of salt, and relative salinity Δc with respect to critical salinity at saturation, has been proposed. The saturation of crystal growth rate is shown to be a dynamic equilibrium between advection and diffusion processes. We show that the stable polygonal geometry of the salt playas is an effort toward the total minimization of system energy.

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Combinatorial topology and geometry of fracture networks

Cited by 7 publications

References 71 publications

Evolution of Tiling-like Crack Patterns in Maturing Columnar Joints

Evolution of Tiling-like Crack Patterns in Maturing Columnar Joints

Evolution of polygonal crack patterns in mud when subjected to repeated wetting–drying cycles

Three-Dimensional Modeling of Polygonal Ridges in Salt Playas

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