Solving the Lamé problem in the quasi-brittle setting to substantiate the solution scheme for rock breaking by the blast-hole method

Introduction. The blast-hole method of mining and preparing rocks for excavation has been used by mankind since ancient times. However, there are no generally accepted and generally recognized design and calculation methods for this process, which was established in the review part of this work. Research objective is to develop a basic analytical model for hard rock quasi-brittle materials breaking by a blast-hole method using static thrusts. Methods of research. A detailed analysis of the breaking mechanism through the transition state from the rock material continuity to its separation is the methodological basis of the developed quasi-static model for calculating the energy-power indicators of the process of separating volumes (pieces) of rock into separate parts. It makes it possible to justify and determine: the irreversible energy of rupture through the energy constant rock material in relation to the process energy efficiency, critical load resulting in local breaking, the size and number of the primordial tension cracks, as well as the number of through cracks. Result. The obtained indicators and parameters formed the body of the energy balance equation of the irreversible energy of rupture of rock pieces. The developed methodology for setting and solving the problem of the limiting state of quasi-brittle materials rupture in the form of a thickwalled cylinder (the Lamé problem) that simulates a certain solid volume deformed by a uniformly distributed blast-hole pressure of thrusts, is universal and can serve as a base for assessing a set of issues related to mining. The resulting interrelation of indicators made it possible to derive an equation for the energy balance of the rock volumes rupture. Due to the solution for this equation, the critical value of the blast-hole pressure that results in local breaking, the size of primordial cracks, and the corresponding limiting burst pressure were obtained for the first time. Theoretical novelty. Within the framework of the general approach of the quasi-brittle materials breaking linear theory, based on the developed methodology, the problem of the limiting state of the rock in the form of natural joint loaded with a static thrust through a single blast-hole, was correctly stated and solved by the energy method for the first time. As a result of the research, an analytical dependence of the rock static breaking energy intensity was obtained, which includes both the volumetric specific energy of potential deformation and the specific surface energy