Geometrical features of crack network formation on rock surfaces during cooling

Oksana Livak*, Serhii Minieiev, Valentyn Hankevych, Iryna Matsiuk

oksana.livak.2019@gmail.com

Received 16.01.2026, Revised 26.05.2026, Accepted 30.06.2026, Published 06.07.2026
Abstract

This study aimed to determine the most probable geometric shape of cracking cells based on the principle of minimising the energy required to cause failure in the surface layer of rocks. The research methodology was based on analysing the classical isoperimetric problem. This states that, among shapes with the same perimeter, the circle has the largest area, while among polygons, the regular n-gon has the largest area. The energetically most favourable configuration of the crack system was substantiated, taking into account the physical conditions governing crack formation. The theoretical propositions were then compared with experimental data and the results of previous thermomechanical rock failure studies. It was found that cooling heated surfaces predominantly results in a polygonal crack structure with hexagonal cells forming. It was demonstrated that the hexagon is the most optimal geometrical figure, in that it allows for dense packing on a plane without the formation of additional voids or stresses, and at the same time provides the minimum perimeter for a given area. The size of the cracking cells was found to be determined by a combination of factors, the most important of which were the physical and mechanical properties of the rock, the magnitude of the temperature gradient and the cooling regime. The characteristic cell size decreases as the temperature difference between heating and cooling increases. The broad spectrum of structural dimensions formed, ranging from fractions of a millimetre to several metres, was confirmed. The practical significance of these results lies in their potential application in predicting fracture parameters and controlling rock failure processes during mineral deposit development, and in designing technologies involving thermal effects on rock masses

Keywords:
thermal cyclic loading; isoperimetric problem; regular tessellation; honeycomb structure; thermal cyclic failure

Retrieved from Vol. 60, No. 1, 2026

Pages 22–33

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Livak, O., Minieiev, S., Hankevych, V., & Matsiuk, I. (2026). Geometrical features of crack network formation on rock surfaces during cooling. Mining Journal of Kryvyi Rih National University, 60(1), 22-33. https://doi.org/10.31721/2306-5435-2026-1-22-33
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