## 1 Introduction
The behavior of soil under very small strains (\(\gamma<10^{-5}\)) is linear elastic, but under higher strains, its behavior is nonlinear. Therefore, it is essential to select an appropriate constitutive model to consider nonlinear strain-dependent stiffness under different loads in a numerical analysis. Although many scholars have used numerical simulation methods to investigate the seismic behavior of soil-structure interactions (e.g., soil-tunnel interactions), an appropriate and more advanced constructive model would better predict the actual behavior of soil under earthquake forces. The use of some constitutive models such as [PERSON]'s multi-yield model and the intergranular strain anisotropy model (ISA) requires complex calibration to determine their various parameters, which decreases their utility in research [23]. However, isotropic/kinematic plastic hardening models, comprising isotropic and kinematic hardening components, accurately predict soil behavior under dynamic loads. The main theories on the hardening of soil are isotropic hardening, the linear and nonlinear kinematic hardening model of Armstrong-Frederick [24], and the nonlinear kinematic hardening model of Chaboche [25]. Isotropic hardening occurs when a yield surface is uniformly large, whereas kinematic hardening occurs when the center of the yield surface moves in stress space without the initial yield surface size changing, which is known as the Bauschinger effect [26; 27]. Finally, if the size and center of the yield surface change simultaneously, this is denoted as combined hardening. As soils exhibit anisotropic kinematic hardening behavior, isotropic models are not suitable for studying soil behavior under seismic loads. Thus, we use a nonlinear kinematic hardening model in this study, which is a simplified version of the combined isotropic/kinematic hardening model generated by assuming that the size of the yield surface is constant [28]. In this study, a kinematic hardening constitutive model based on the Von [PERSON] failure criterion is been used to investigate the seismic behavior of tunnels in clay soil. In some constitutive models, such as the Cam-clay and modified Cam-clay models used in previous studies, the yield surface is presumed to expand with increasing loading level. This means that when the soil experiences a larger stress surface under dynamic loading, the behavior of the materials will be quite elastic at lower loading levels until the soil sample again experiences stresses greater than the previous yield stress. Therefore, these constitutive models cannot model the Bauschinger effect nor determine the hysteresis loops that exist under cyclic loading, which means that these models cannot predict the actual behavior of soil under kinematic loads, such as seismic loads. However, a kinematic hardening model can be used as an advanced model for the simulation of soil behavior, thereby overcoming these problems. Here, the phenomenon of liquefaction has not been considered and there is no generation of pore pressures. The phenomenon of cyclical degradation and degradation soil stiffness are due to soil structure. We investigate the factors affecting the behavior of tunnels in clay under earthquake load, including the earthquake intensity, tunnel shape, embedment depth, and lining thickness. Then, we compare our results with the results of [PERSON] et al. [29]. The earthquake used in this study is a 1D earthquake with a dominant frequency of 1 Hz, and the 0.4 g and 0.83 g maximum accelerations that occurred in 1995 in Kobe, Japan. A viscous spring boundary is used around the soil model to prevent the return of earthquake waves into the model, thus ensuring that actual results are obtained.