PHYSICAL REVIEW E 105, 034801 (2022) Combinatorial topology and geometry of fracture networks A. Roy , 1, 2, 3 R. A. I. Haque , 2, 3 A. J. Mitra , 4 S. Tarafdar , 3 and T. Dutta 2, 3, * 1 Physics Department, Charuchandra College, Kolkata 700029, India 2 Physics Department, St. Xavier’s College, Kolkata 700016, India 3 Condensed Matter Physics Research Centre, Jadavpur University, Kolkata 700032, India 4 Mathematical Sciences, Montana Tech, Butte, Montana 59701, USA (Received 1 August 2021; accepted 21 February 2022; published 10 March 2022) A map is proposed from the space of planar surface fracture networks to a four-parameter mathematical space, summarizing the average topological connectivity and geometrical properties of a network idealized as a convex polygonal mesh. The four parameters are identified as the average number of nodes and edges, the angular defect with respect to regular polygons, and the isoperimetric ratio. The map serves as a low-dimensional signature of the fracture network and is visually presented as a pair of three-dimensional graphs. A systematic study is made of a wide collection of real crack networks for various materials, collected from different sources. To identify the characteristics of the real materials, several well-known mathematical models of convex polygonal networks are presented and worked out. These geometric models may correspond to different physical fracturing processes. The proposed map is shown to be discriminative, and the points corresponding to materials of similar properties are found to form closely spaced groups in the parameter space. Results for the real and simulated systems are compared in an attempt to identify crack networks of unknown materials. DOI: 10.1103/PhysRevE.105.034801 I. INTRODUCTION A large body of literature exists on the formation and propagation of cracks and the network of structures they form [1–8]. Fluid transport through a “crack network” [9–12], scal- ing laws in fracture interfaces [13–16], and mechanisms of failure [17–20] are certain aspects of fracture systems that have been widely studied since the early 1980s. Some inter- esting mathematical aspects of the crack networks, however, remain inadequately focused. In this paper our concern is not a single crack but a network of cracks. Cracks often form networks with distinctive patterns, as seen for example, in mud cracks besides a dried river bed or in a shattered glass pane. The patterns consist of solid polygonal shapes separated by narrow gaps, which are the cracks. Borrowing a term from geology we may call the solid polygons “peds.” The focus of our interest is on two aspects: (i) the geometry, that is, the shapes and relative sizes (size distribution) of the peds and how they depend on various factors creating the crack pattern, and (ii) the topology, that is, the connectivity of the pattern— how the adjacent peds connect with each other and how the crack network pervades the whole system and related features. Our goal is to collect experimental results on real systems as well as crack networks simulated through different algorithms to finally analyze and classify them using the above concepts and appropriate measures. Various statistical measures have often been used to de- scribe spatial structures in physical systems, porous systems, complex fluids, and biological and cosmological systems * Tapati Dutta tapati_mithu@yahoo.com [21–26]. The scale-invariant branching structure of the crack network makes it a natural paradigm for fractal systems and several such. studies exist [27–32]. Summary statistics such as distance characteristics, spherical contact distribution func- tion, and J function, as well as second-order characteristics such as two-point correlation function, Ripley’s K function [33], and the L function have generally been utilized [34–39]. Mecke et al. [40] introduced a morphological description of a triplet function that constituted of normalized values of integral-geometric quantities of area, boundary length, and Euler number of patterns of disks centered on the points of a stationary point distribution, in an approach similar to Adler [41] and Worsley [42–44]. Andresen et al. [45] have analyzed the topology of three-dimensional fractured systems as an abstract map of nodes and links using tools of network theory. Hope et al. [46] have worked with Poissonian dis- crete fracture model and a mechanical discrete fracture model in three dimensions (3D) to study the effect of constrained fracture growth models on topology. However, a systematic and comprehensive topology-geometry based study of planar polygonal crack networks where the physical crack network is considered as a tessellation of the Euclidean plane seems to be missing. As planar polygonal tessellations (also known as mosaics or tilings) are a vibrant area of geometry, it would be useful to bring forth established tools and measures from geometry to the study of physical crack networks, which are often polygo- nal or nearly polygonal. In the present paper, we propose a 4-tuple (n, v, D,λ) to classify planar surface crack networks, including both convex and nonconvex polygons, as discussed in detail in Sec. II. In brief, the first two elements of the 4-tuple are the average 2470-0045/2022/105(3)/034801(15) 034801-1 ©2022 American Physical Society