## 2 Problem Statement  
This study aims to study the interaction between the surrounding rock and the lining subjected to inner surface pressure in the radial direction under hydrostatic pressure. The plan sketch of the calculation model is shown in Figure 1. The assumptions of this study are the following:  
1. The horizontal and vertical pressure are equal. 2. The rock mass is a homogeneous, isotropic, and viscoelastic material [10]. 3. The excavation radius of the circular tunnel is \(R_{2}\). 4. The lining is an inhomogeneous, isotropic, and elastic material. 5. The inner and outer radii of the lining are \(R_{1}\) and \(R_{2}\), respectively. 6. The thickness of the lining is \(d\). 7. The mechanical property (such as elastic modulus) of the lining varies only along the radial direction as a power function. 8. The inner surface of the lining is subjected to the pressure \(q(t)\) in the radial direction. The Burgers model [37] is selected to describe the rheological behavior of the rock mass in this paper. The Burgers model is a comprehensive model that includes both the Maxwell and Kelvin models in series, as shown in Figure 2. \(G_{\rm m}\) and \(\eta_{\rm m}\) are the shear modulus and viscosity of the Maxwell model, respectively; \(G_{\rm k}\) and \(\eta_{\rm k}\) are the shear modulus and viscosity of the Kelvin model, respectively; and \(t_{\rm m}\) and \(t_{\rm k}\) are the relaxation and retardation times of the Maxwell and Kelvin models, respectively. When the dashpot element of the Maxwell model is ignored (\(\eta_{\rm m}\to+\infty\)), the Burgers model is transformed to the Kelvin-Voigt model. When both spring and dashpot elements of the Maxwell model are ignored (\(G_{\rm m}/\eta_{\rm m}\to+\infty\)), the Burgers model is transformed to the Kelvin model. When both spring and dashpot elements of the Kelvin model are ignored (\(G_{\rm k}/\eta_{\rm k}\to+\infty\)), the Burgers model is transformed to the Maxwell model. The behaviors of the initial instantaneous strain, subsequent transient creep, and final steady creep of the Burgers model are controlled by the spring and dashpot elements of the Maxwell model, and the spring and dashpot elements of the Kelvin model [38]. Figure 1: Plan sketch of the calculation model. Figure 2: The Burgers model and its transformations. The volumetric deformation of the rock is only elastic dilation under hydrostatic stress, and its rheological behavior is mainly affected by the deviatoric stress. Therefore, the stress and strain of the rock can be decomposed as follows:  
\[\left\{ \begin{array}{l}\sigma_{ij}^{\mathrm{R}}=\delta_{ij}\frac{\sigma_{ik}^{ \mathrm{R}}}{\frac{\sigma_{ik}^{\mathrm{R}}}{3}}+s_{ij}^{\mathrm{R}}\\ \varepsilon_{ij}^{\mathrm{R}}=\delta_{ij}\frac{\varepsilon_{ik}^{\mathrm{R}}} {\frac{\varepsilon_{ik}^{\mathrm{R}}}{3}}+\varepsilon_{ij}^{\mathrm{R}}\end{array} \right. \tag{1}\]  
where \(\sigma_{ij}^{\mathrm{R}}\) and \(\varepsilon_{ij}^{\mathrm{R}}\) are the stress and strain tensors of the rock, respectively; \(s_{ij}^{\mathrm{R}}\) and \(\varepsilon_{ij}^{\mathrm{R}}\) are the deviatoric stress and strain tensors of the rock, respectively; \(\sigma_{ik}^{\mathrm{R}}\) and \(\varepsilon_{ik}^{\mathrm{R}}\) are the volumetric stress and strain of the rock, respectively; and \(\delta_{ij}\) is the Kronecker delta. The constitutive equation of the integral form of the Burgers model can be expressed as follows:  
\[\left\{ \begin{array}{l}\varepsilon_{ij}^{\mathrm{R}}=J(t)*d\delta_{ij}^{ \mathrm{R}}\\ I(t)=\left[\frac{1}{2\Omega_{m}}+\frac{t}{2\eta_{m}}+\frac{1}{2\Omega_{ \mathrm{c}_{k}}}\big{(}1-e^{-t/t_{\mathrm{k}}}\big{)}\right]\\ \varepsilon_{ii}^{\mathrm{R}}=\sigma_{ii}^{\mathrm{R}}/3K^{\mathrm{R}}\end{array} \right. \tag{2}\]  
where \(J(t)\) is the creep compliance of the Burgers model; the asterisk \"*\" denotes the convolution algorithm; and \(K^{\mathrm{R}}\) is the bulk modulus of the rock.