## 1 Introduction  
Tunnel engineering is one of the primary infrastructure projects in the road, railway, rail traffic, mining engineering, and military sectors [1; 2; 3; 4; 5]. Tunnel construction inevitably affects the stress state of the surrounding rock. Tunnel lining, one of the major support structures of a tunnel project, plays a crucial role in the tunnel's whole life [6; 7; 8]. The stability and safety of a tunnel in service are jointly controlled by the conditions of surrounding rock and lining. Therefore, research on rock-lining interaction is of great significance for evaluating the tunnel's safety throughout its lifespan. The changes in a tunnel wall's deformation over time are closely linked to the time-dependent characteristics of the surrounding rock caused by tunnel excavation [9; 10; 11]. The time-dependent characteristics of the rock, also named its rheological behavior, have been thoroughly investigated by a series of rheological experiments on soft rock. The results showed that 30-70% of the total deformation is caused by the rheological behavior of the rock [12; 13; 14]. Additionally, due to the rheological behavior of the rock mass, the deformation of the surrounding rock and the pressure acting on the lining increase with service time, as observed in many actual projects such as the Ureshino tunnel [15] in Japan, the Lyon-Torino Base tunnel [16] in Italy and France, the Shibli tunnel [17] in Iran, Baijiao coal mine [3], the Minxian tunnel [18], and the Muzhailing tunnel [19] in China. Researchers have studied the rock-lining interaction and considered the surrounding rock's rheological behavior using various methods, such as field monitoring [20; 21], model experiments [22; 23], numerical simulations [24; 25], and theoretical analyses [26; 27]. Tunnel excavation is a three-dimensional problem, especially around the tunnel cutting face. To simplify this problem in the theoretical analysis, the concept of fictitious support pressure has been proposed by many researchers [28; 29; 30]. After the fictitious support pressure is imposed on the tunnel wall, the three-dimensional problem of the tunnel face can be transformed into a two-dimensional one [31; 32], allowing for easier analytical solutions. These solutions take into account the mechanical responses of the rock-lining interface during excavation and consider different rheological models. For example, [PERSON] et al. [10] presented analytical models to investigate the mechanical behavior of the rock-lining interface during tunnel excavation, taking into account both the tunnel face effect and the rheological behavior of the rock mass. [PERSON] [33] deduced a viscous elastic-plastic solution to explore the stress-displacement distribution around unlined and lined tunnels, considering the tunnel face effect. However, there are limited studies on how the lining affects the rock-lining interaction after installation when subjected to radially inner surface pressure. In tunnel projects, such as water and high-speed railway tunnels, the lining is subjected to inner surface pressures in the radial direction after tunnel construction. Based on the complex variable method, closed-form solutions have been proposed to explore the stress-displacement field around a circular tunnel whose lining is under hydrostatic pressure and thus under a radially inner surface pressure. However, the rheological behavior of the rock mass is not considered in these solutions [34; 35; 36]. When analyzing the rock-lining interaction, the lining is typically assumed to consist of a homogeneous and isotropic material. This assumption, which is incompatible with the actual conditions, certainly affects the analytical results of the rock-lining interaction.