## 1 Introduction  
Traditional anchor rod and mesh support systems significantly impact the tunneling speed in coal mines because of their long installation periods [1, 2]. An efficient and fast method for temporary tunnel support is the spraying support technology [3, 4, 5, 6], which involves injecting liquid materials into the surrounding rock through the surface for attachment and curing, thereby improving its mechanical properties. This technology, characterized by high mechanization, can effectively stabilize the surrounding rock in areas such as the overhead and sidewalls of the tunneling face. The technology also provides space and time for anchor installation, thus enhancing tunneling speed and improving the coordination of the mechanized excavation system [7, 8].  
Polymer spraying materials can substantially increase the strength of surrounding rock within a short period. Globally, researchers have conducted extensive experiments on coal rock bodies after applying polymer spraying materials, examining the tensile, compressive, and shear strengths. These studies have indicated that such sprayed coatings can effectively improve the mechanical properties of coal rock bodies [9, 10, 11, 12, 13, 14]. The curing time of polymer coatings is much shorter than that of concrete (in the order of seconds, approximately 10 s) [15]; the time to reach peak strength is also shorter [16]. When polymer materials are sprayed onto the surface of a surrounding rock, they form a continuous film that effectively prevents swelling, loosening, and slippage of the surrounding rock. They force interactions between the rock particles and provide a self-stabilizing effect, thus replacing the role of metallic meshes [9; 17]. Additionally, owing to their excellent air-tightness, polymer coatings can block air from entering fissures, preventing the crack propagation and weathering of the surrounding rock [18]. Moreover, as a flexible material, the polymer coating applies a flexible confining pressure on the surrounding rock, altering the bearing mechanical characteristics of the rock [19] and demonstrating certain energy-absorption capabilities [17; 20].  
However, the interaction between the coating and surrounding rock remains unclear, particularly the plastic deformation of the surrounding rock under the confining pressure of the coating when the coal pillar becomes brittle through mining activities. This unclear aspect requires further research and the establishment of theoretical models. Furthermore, the thickness of polymer coatings typically used in engineering applications is on the millimeter scale [21; 22; 23]; however, the coating thickness still needs further optimization.  
In this study, we conducted uniaxial compression tests on coal before and after the spraying of polyurea [24; 25], comparing the failure modes and stress-strain curves. An elastic-theory-based stress-strain model for coal under polyurea spraying conditions was established and experimentally validated. The study revealed the deformation and failure patterns of coal under polyurea spraying, clarifying how the coating applies confining pressure to maintain residual strength in coal after brittle failure. The model was also used to compare the residual strength of coal with different coating thicknesses.