In various fields of mathematical research, the Brouwer degree is a potent tool for topological analysis. By using the Brouwer degree defined in one-dimensional space, we interpret the equation of state for temperature in black hole thermodynamics, $$T=T(V,x_i)$$ T = T ( V , x i ) , as a spinodal curve, with its derivative defining a new function f. The sign of the slope of f indicates the topological charge of the black hole’s critical points, and the total topological charge can be deduced from the asymptotic behavior of the function f. We analyze a spherical hairy black hole within the framework of Lovelock gravity, paying particular attention to the topological structure of black hole thermodynamics under Gauss–Bonnet gravity. Here, the sign of the scalar hair parameter influences the topological classification of uncharged black holes. When exploring the thermodynamic topological properties of hairy black holes under cubic Lovelock gravity, we find that the spherical hairy black hole reproduces the thermodynamic topological classification results seen under Gauss–Bonnet gravity.

Fracture within the rock is one of the main factors leading to rock destabilization and has a significant effect on the stability of the project. In this study, sandstone is used as a research target, specimens with crack inclination angles of 0°, 30°, 45°, 60°, and 90° are prefabricated, and the split Hopkinson pressure bar (SHPB) impact test of sandstone with cracks is carried out based on digital image recognition technology to explore the dynamic damage characteristics of the specimens with five angles. The basic mechanical parameters of sandstone are tested to determine the RHT model intrinsic parameters, and the numerical computational RHT model of sandstone containing crack defects is established, which is verified in comparison with the test to analyze the validity of the model. Finally, the failure characteristics of the numerical model under initial stress were carried out. The study shows the following: with the increase in the fracture angle, the dynamic compressive strength and deformation modulus are distributed in a slanting V-shape, and the inclination angle of 45° is the smallest. The strain rate and energy dissipation rate are distributed in a slanting N-shape, and the inclination angle of 45° is the largest. The transmittance shows a decreasing trend, which is the opposite of the reflectivity pattern. The crack angle determines the location and direction of the initial crack, which affects the failure mode. In addition, the parameters of the RHT constitutive model suitable for sandstone are obtained, and the damage and strength patterns of the established RHT model are highly consistent with the laboratory test results. The damage range of numerical models for crack defects with different inclination angles is negatively correlated with confining pressure values and positively correlated with axial pressure values. The damage zones are symmetrically distributed approximately perpendicular to the direction of cracks, and the confining pressure has a contributing role in the peak of the element stresses; however, the axial compression has no contribution in the peak of the element stresses.

This work explores the universal topological classes of various black hole configurations immersed in a quintessence field. The results indicate that the Schwarzschild black hole surrounded by quintessence is of the $$W^{1-}$$ W 1 - type, exhibiting thermodynamic instability in both extremal temperature limits. The introduction of rotation alters its topological nature to the $$W^{0+}$$ W 0 + class, where the Kerr black hole displays coexistence of a stable small black hole and an unstable large black hole at low temperatures. In comparison, anti de Sitter (adS) black holes display different thermodynamic topologies. The Schwarzschild–adS solution corresponds to the $$W^{0-}$$ W 0 - class, where an unstable small branch and a stable large branch emerge in the high temperatures. Meanwhile, the Kerr–adS classified as $$W^{1+},$$ W 1 + , maintains stability in both high- and low-temperature limits. Furthermore, the 4D/5D quintessential black hole belongs to the $$W^{0+}$$ W 0 + class within the Einstein–Gauss–Bonnet framework. Overall, parameters such as rotation and Gauss–Bonnet in quintessence background field affect the topological classifications of the black holes, while the quintessence field primarily modifies the stability range without changing the topological class itself.

To address the challenges of inconvenient material sourcing and low stemming efficiency in horizontal tunnel blasting, this study proposes a novel rapid-setting gun mud. The material’s composition, formation mechanism, and initial setting characteristics are first analyzed, and a multi-parameter coupled model for optimal stemming length is established. Subsequently, a series of blasting model tests with varying stemming lengths is conducted, followed by a quantitative analysis of the block-size distribution of limestone under different stemming conditions. The influence of stemming length on rock fragmentation characteristics is revealed, and theoretical predictions are validated against experimental results. Findings indicate that the curing process of the proposed material is primarily driven by hydration reactions, with simple operation, environmental friendliness, and safety for workers. Moreover, the developed multi-parameter coupled model for optimal stemming length shows good agreement with experimental results, providing a quantifiable theoretical basis for the refined parameter design in mechanized tunnel blasting.

In modern tunneling construction, the blasting seismic effect seriously threatens the safety and stability of the existing engineering construction. Meanwhile, due to the complexity of blasting loading, there is a very large difference from real blasting by using the sample triangle dynamic loading model or trapezoid dynamic loading model to analyze the blasting vibration problem. In this paper, based on the analysis of the pressure change, volume expansion, fracture development, and blasting gas motion, an accurate blasting loading model was proposed. Then, adopting plane longitudinal shock theory, the coupled loading of multiple holes can be obtained. Subsequently, the 3D simulation analysis of the tunnel blasting shows an error of 2% compared with on-site monitoring data, meeting engineering requirements. Finally, based on the simulation results, regression prediction function of the velocity curve and effective tensile stress curve under different safety criteria are established, achieving accurate prediction of the damage degree of the host rock and liner line under different engineering requirements.