## 1 Introduction  
Underground workings for compressed air energy storage (CAES) with rock linings typically employ one of three sealing systems: steel-lined sealing layers [1], flexible sealing layers using polymer materials [2, 3], and polyurethane polymer mortar [4]. Compared to steel linings, flexible sealing layers offer greater deformability, faster and simpler construction, and lower costs, making them a promising alternative [5]. However, under high internal gas pressure, linings may crack, potentially leading to the failure of the flexible sealing layer due to crack intrusion [6, 7, 8, 9]. As a result, flexible-sealed CAES workings impose strict limits on the maximum allowable crack width in the lining. According to the Chinese Design Specification for Hydraulic Concrete Structures [10], to prevent reinforcement corrosion, the maximum permissible crack width for reinforced concrete structures permanently located underground is 0.3 mm. This requirement is difficult to meet under conditions of high internal pressure and lower-quality rock mass. When the rock mass has a low elastic modulus, significant deformation can lead to wide cracks in the lining [11]. Consequently, monolithic cast-reinforced concrete linings are generally applicable only when the surrounding rock is of higher quality or the storage pressure is relatively low. Designing high-pressure CAES workings in rock masses of lower quality presents significant challenges. In addition, CAES workings are required to withstand high internal pressure, which demands both high gas tightness and structural stability [12; 13; 14]. [PERSON] et al. [15] and [PERSON] et al. [16] developed a thermo-mechanical coupled numerical model for CAES workings, showing that tensile stresses during operation can induce cracking in the lining and surrounding rock, thereby compromising overall stability. To accommodate the surrounding rock deformation and reduce cracking in the lining, several strategies have been proposed. One approach is to use flexible concrete materials with a low elastic modulus and high tensile strength for the lining [17]. Since the load-bearing ratio of monolithic linings increases with the elastic modulus [18], selecting a lower-modulus material allows the lining to act mainly as a load-transfer structure. This reduces the risk of cracking [19; 20]. Another approach is to adopt a segmented lining structure that allows the radial displacement to be released through preset seams, thus preventing wide cracks. The earliest segmented lining consists of eight reinforced concrete segments and backfill concrete between the segment tails and the surrounding rock [21; 22; 23]. Segmented linings can move with rock deformation, and the internal gas pressure is transferred through the sealing system to the surrounding rock. Field tests have shown an air leakage rate of only 0.2% per day [24], indicating that segmented linings can meet sealing requirements and are suitable for CAES working construction. [PERSON] et al. [25] proposed an analytical model to determine the support stiffness of circumferentially yielding linings, which was successfully applied in the Saint-Martin-la-Porte tunnel. Their simulations demonstrated that yielding linings, formed by embedding yield elements in shotcrete to create pre-designed seams, effectively transfer pressure to the rock mass through deformable seams and reduce structural loads. [PERSON] et al. [26] proposed a segmented lining structure for CAES workings based on elastic deformation analysis, though it did not account for plastic deformation. [PERSON] et al. [27] introduced a composite segmented lining. This design considered shear deformation between the initial support and the lining. However, the shear properties of the sliding layer were not validated through experiments. In addition, the effects of the lateral pressure coefficient, the number of seams, and their positions were not considered. [PERSON] et al.