## 2 Basic Theory and Realization Method of SSD Coupling  
### Basic Theory of SSD Coupling  
Changes in the seepage-stress coupling environment can cause changes in the internal microstructure (i.e., meso-damage) [30], macro-mechanical properties, and permeability of concrete. Changes in permeability and mechanical properties can affect the concrete'sstress state, the distribution of soil pore pressure and worsen the meso-damage of concrete. This phenomenon is called SSD coupling. Conventional seepage-stress coupled governing equations include solid-based geometric equations and equilibrium equations, fluid-based mass conservation equations and flow equations, and seepage-stress coupled constitutive equations [31]. The SSD coupled governing equations can be obtained by introducing concrete damage variables into the conventional seepage-stress coupled equations. The following derivation techniques are explained by using direct tensor notation to simplify the theoretical formulations mathematically [32]. Assuming that the seepage process follows the nonlinear Darcy's law in the entire section, water and materials are incompressible, and the volumetric deformation of the saturated porous solid framework is equal to the deformation of the pores, then, the seepage field conforms to the continuity equation of three-dimensional single-phase porous fluid [33; 34]. \[\frac{\partial}{\partial x}\bigg{[}\mathbf{k}_{x}\frac{\partial H}{\partial x} \bigg{]}+\frac{\partial}{\partial y}\bigg{[}\mathbf{k}_{y}\frac{\partial H}{ \partial y}\bigg{]}+\frac{\partial}{\partial z}\bigg{[}\mathbf{k}_{z}\frac{\partial H }{\partial z}\bigg{]}+\mathbf{Q}=\mathbf{0} \tag{1}\]  
where \(\mathbf{k}_{x}\), \(\mathbf{k}_{y}\) and \(\mathbf{k}_{z}\) are the permeability coefficients in the \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\) directions, respectively; hydraulic potential \(\mathbf{H}=\frac{\mathbf{p}}{\gamma}\), in which \(\mathbf{p}\) is the pore water pressure and \(\mathbf{\gamma}\) is the water unit weight; \(\mathbf{z}\) is the elevation head; and \(\mathbf{Q}\) is the source sink term. Assuming that concrete and rock masses are equivalent continuum models, then, after finite element discretization, interpolation, and integration, the matrix equation for solving the seepage field can be obtained as follows [35]:  
\[[\mathbf{A}]\{\mathbf{H}\}=\{\mathbf{F}\} \tag{2}\]  
where \([\mathbf{A}]\) is the total permeability matrix, \(\{\mathbf{H}\}\) is the column vector of the node head, and \(\{\mathbf{F}\}\) is the nodal load obtained by integrating the seepage boundary. After the seepage field is calculated, the water load generated by the hydraulic gradient acts on the inside of the structure in the form of seepage force. In the equivalent continuum model, the seepage gradient acts on the node in the form of seepage force. After the node head is obtained through the seepage field calculation, the seepage load acting on the element node is calculated as follows:  
\[\{\mathbf{F}_{\mathbf{p}}\}=-\int\int\limits_{\mathbf{\Omega}}\int\mathbf{\gamma}[\mathbf{N}]^{T} \bigg{\{}\frac{\partial\mathbf{H}}{\partial\mathbf{x}},\frac{\partial\mathbf{H}}{\partial y },\frac{\partial\mathbf{H}}{\partial z}-\mathbf{1}\bigg{\}}^{T}\mathbf{d}\mathbf{\Omega} \tag{3}\]  
where \([\mathbf{N}]\) is the interpolation function, \(\mathbf{\Omega}\) is the integral domain of the seepage force node and \(\mathbf{T}\) is the transposition of a matrix.