## 1 Introduction  
The development of the pressure of a rock mass that acts on an excavation support depends on various factors, including the physical and mechanical properties of the rock mass, the mining technology used, and the characteristics of the support [1]. As the excavation face advances, the rock mass gradually moves toward the excavation axis. The increment of the displacement of the rock mass is influenced by both the distance to the excavation face and the time. However, in practical applications, the influence of time is often overlooked [2]. An analysis of the impact of the time factor on the growth of the displacement of rock masses requires the utilization of various rheological models--as discussed, for example, in [3; 4; 5; 6; 7].  
The development of a deformational pressure acting on an excavation support is the result of the suppression of the increase in the displacement of the rock mass increment caused by the installation of the support [1]. The magnitude of the deformational pressure acting on a support system decreases as the ground displacement increases. Consequently, its value is a function of the distance between the excavation face and the support location during installation [2; 8; 9].  
Up to a certain level of ground displacement, the rock mass in the vicinity of an excavation acts as an elastic medium. Once the displacement of the ground exceeds a certain threshold, inelastic zones begin to form around the excavation, which could lead to a reduction in the rock mass within these zones [4; 10]. Therefore, four different types of inelastic zones can be identified:  
* The compressive strength of the rock mass within the inelastic zone is equal to the compressive strength in the elastic zone, satisfying the condition: \[R_{cg}=R^{{}^{\prime}}_{cg}\] (1) This scenario characterizes the elasto-perfectly plastic medium.
* The compressive strength of the rock mass within the inelastic zone is lower than the compressive strength in the elastic zone, meeting the following condition: \[R_{cg}>R^{{}^{\prime}}_{cg}\] (2) This represents the softening elasto-plastic medium.
* The rock mass within the inelastic zone exhibits no compressive strength: \[R^{{}^{\prime}}_{cg}=0\] (3) This represents the elasto-fractured medium.
* Two coaxial inelastic zones form around the excavation. In the immediate vicinity of the excavation, there exists a fractured rock mass zone without compressive strength. At a certain distance from the excavation boundary, a zone of plastic rock mass develops, retaining some residual compressive strength. This scenario denotes an elasto-plasto-fractured medium [11].  
The development of inelastic zones around the excavation corresponds to the occurrence of a static rock mass pressure, which arises from the gravitational load of plasticized and fractured roof rocks. As the value of the ground displacement increases, the extent of the inelastic zones, and thus the level of static pressure, also increases. The elasto-plastic rock mass exhibits a boundary displacement value at which the static pressure reaches its maximum, while the deformational pressure diminishes. Therefore, the elasto-plastic model produces a paradox, since it suggests that the value of the support pressure always decreases along with an increase in ground displacement, whereas according to practical engineering, the support pressure is expected to increase significantly after reaching a certain value of ground displacement as a result of the loss of confinement and the detrimental loosening of the rock mass.  
Meanwhile, in elasto-fractured and elasto-plasto-fractured media, as displacements increase, the static pressure theoretically tends toward infinity, while the deformational pressure approaches zero. Therefore, the final value of the support pressure increases after reaching a certain value of ground displacement, which seems to be consistent with engineering practice [12; 13].