## 1 Introduction  
In recent years, more and more tunnels in high-altitude and high-latitude areas have been constructed in China, thus the problem of frost damage in old tunnels in cold regions is becoming more and more prominent [1]. Apart from icing problems, several tunnels with a service life of more than 10 years have demonstrated the phenomena of serious cracking of the lining, brittle lining, partial peeling, and falling off near the portal [2]. In contrast to the problem of icing in tunnels, the deterioration of the lining concrete affects the bearing structure, which will seriously threaten the safety of the tunnel structure. Additionally, the problem of the lining blocks falling off will have an even more serious impact on traffic safety than will the falling of ice. Therefore, it is very important to research the rules and influence of tunnel lining material deterioration in cold areas to increase our knowledge of maintenance and operation safety. At present, research on the frost damage of tunnels in cold regions mainly focuses on the temperature field of the tunnel and the influence of frost damage. The analytical calculation method, model experiments, and in situ tests are the main methods used in these studies. A series of analytical solutions considering different boundary conditions, different parameters, and different models were carried out to study the frost heaving force in cold-region tunnels. Based on the initial elastic state of the surrounding rock, a semi-analytical method to calculate the stress in the plastic was proposed [3]. Then, the entire surrounding rock was divided into four zones: the unfrozen elastic zone, the frozen elastic zone, the frozen plastic zone, and the support zone. In addition, a more complicated elastoplastic calculation model for the surrounding rock in cold-region tunnels was established [4]. Meanwhile, based on the stochastic analysis model and the stochastic finite element method, the stochastic mechanical characteristics of tunnels in cold regions were investigated [5]. Furthermore, an analytical elastoplastic solution for frost having force was recently proposed, in which an anisotropic frost heave coefficient, k, was introduced to consider transversely isotropic frost heaving [6]. Then, a novel noncircular solution for frost heaving force was established and verified using the example of the Zhegu Mountain tunnel [7]. Additionally, aiming at the prediction of crack initiation and propagation in brittle materials, an efficient parameter estimation framework was proposed, involving a step-wise Bayesian inversion framework for ductile fractures to provide accurate knowledge regarding the effective mechanical parameters [8; 9]. These numerical calculation methods for the frost heaving force of tunnels include comprehensive explanations and support for the topic of freezing damage and prevention in cold-region tunnels. On the other hand, the analytical calculation and model test were carried out to study the distribution laws and factors of the tunnel temperature field. Based on the analytical results of the temperature field in the cross-section and the longitudinal direction of tunnels [10; 11], a three-dimensional analytical model of the temperature field was proposed [12]. The result show that the freezing length reduced from 143 m to 70 m and the maximum radial freezing depth at the entrance is also shortened from 4.7 m to 2.4 m due to a 5 cm insulation layer. The analytical calculation of the temperature field in the tunnel has achieved a comprehensive research result. However, their analytical calculation models took the temperature function over a period of one year as the boundary condition, but the influence of daily periodic temperature change on freezing length and freezing depth was ignored. As a result, the deterioration of lining structure subjected to daily freeze-thaw cycles was ignored.