## 1 Introduction
Some classical methods for assessing the stability of the protective pillars and edge sections of coal seams include the methods of visual inspection in situ and a number of geophysical methods that allow for the determination of a relative concentration of the abutment pressure and rock-burst potential [12]. Currently, the number of in situ methods have increased due to development of the methods for assessing the degree of pillar stability based on borehole drilling and determining the parameters of the actual load-bearing pillar core [13]. It is well known that the physical and mechanical properties of rocks have high spatial variability even within the same deposit. In this regard, probabilistic methods for assessing possible protective pillar damage have become widespread as an alternative to classical deterministic approaches. The majority of these estimates are based on the Monte-Carlo simulation method (e.g., see [14; 15; 16]). Recently, the methods for predicting the stability of protective pillars based on machine learning algorithms and neural networks have become popular [10; 16; 17; 18; 19]. Such approaches by means of probabilistic modeling allow for the simulation of many different scenarios and, therefore, establishing a database for the effective forecasting of the pillar behavior. Moreover, the possibility of considering several factors simultaneously and recognizing the patterns of rock mass behavior strongly enhances the probabilistic approach. Despite the competition between the deterministic and probabilistic approaches, they both use the UCS of a pillar as a key characteristic for stability analysis. In other words, the limit design seems to be among the main approaches to assessing the pillar stability. However, the probabilistic approach has the advantage of taking into account the spatial variability of rock properties. With the development of high-performance-computing technology, a large number of numerical methods have been involved in pillar stability calculations, e.g., [20]. Recall that numerical modeling enables the consideration of many factors affecting the rock mass behavior. These are gravity forces, lithology, tectonic stresses, natural discontinuities,geotechnical and mining parameters, etc. When the model parameters are carefully calibrated against the experimental and/or in situ observations, the pillar stability assessment is reliably improved. For instance, [PERSON] et al. [20] justified the application of a new system of cables and anchors for roof and rib support based on the numerical modeling of stope face advancement, which successfully solved the problem of improving stability. Another issue that we have to bear in mind is the actual structure of the rock mass and the complexity of the geotechnical conditions. Due to the depletion of shallow seams, many mines surpassed the 1000 m depth in mine operations several years ago, which increased the vertical load of the pillars and edge sections of seams [21]. Moreover, the sufficiently large dip angles of the coal seams might produce equipment-sliding problems and frequent shifts of the abutment pressure maximums towards roadways [22]. The latter results in the worsening of the geotechnical conditions; meanwhile, ground control becomes extremely difficult. An estimation of the pillar stability of inclined seams is a more complex problem than that in the case of flat-dipping seams. This is due to the additional action of tangential stresses causing a redistribution of loads and a change in the geometry of the actual load-bearing core. A good overview of the inclined seam instability problem can be found in [23]. The authors show that the classical formula for calculating \(F_{\mathrm{s}}\) becomes inappropriate in this case. In this work, we apply the finite-difference continuum damage mechanics approach to modeling the stress-strain evolution of the rock mass during the extraction of adjacent inclined longwall panels. We propose a new modification of the damage accumulation kinetic equation in this study.