## 2 Basic Theory and Realization Method of SSD Coupling
### Basic Theory of SSD Coupling
\tag{10}\]  
where \(\dot{\widetilde{\mathbf{\epsilon}}}^{\mathbf{p}\mathbf{I}}_{max}\) is the maximum value the plastic strain rate tensor, \(\dot{\widetilde{\mathbf{\epsilon}}}^{\mathbf{p}\mathbf{I}}_{min}\) is the minimum value of the plastic strain rate tensor, \(\dot{\widetilde{\mathbf{\epsilon}}}^{\mathbf{p}\mathbf{I}}_{t}\) is the equivalent plastic strain rate in tension, and \(\dot{\widetilde{\mathbf{\epsilon}}}^{\mathbf{p}\mathbf{I}}_{c}\) is the equivalent plastic strain rate in compression. The multiaxial stress weighting factor \(\mathbf{r}\big{(}\widehat{\mathbf{\sigma}}\big{)}\) can be defined as follows:  
\[\mathbf{r}\big{(}\widehat{\mathbf{\sigma}}\big{)}=\frac{\sum_{t=1}^{3}\langle\widehat{ \mathbf{\sigma}}_{t}\rangle}{\sum_{t=1}^{3}|\widehat{\mathbf{\sigma}}_{t}|},\mathbf{0}\leq \mathbf{r}\big{(}\widehat{\mathbf{\sigma}}\big{)}\leq\mathbf{1} \tag{11}\]  
where \(\widehat{\mathbf{\sigma}}_{t}\) (\(\mathbf{i}=1\), \(2\), \(3\)) are the principal stress components, respectively, \(\langle\cdot\rangle\) is defined as \(\langle\mathbf{x}\rangle=(|\mathbf{x}|+\mathbf{x})/\mathbf{2}\), and \(|\mathbf{x}|\) is the absolute value of \(\mathbf{x}\). Under periodic alternating loads, the complex concrete damage mechanism is related to the cracking and merging of the initial cracks and their interrelation during changes. When the concrete is subject to compression after tension, its stiffness will be partially restored, that is, the unilateral effect is more significant. To reflect this effect, the relationship between tensile and compressive damage variables, \(\mathbf{d_{t}}\) and \(\mathbf{d_{c}}\) is:  
\[(\mathbf{1}-\mathbf{d})=(\mathbf{1}-\mathbf{s_{t}}\mathbf{d_{c}})(\mathbf{1}-\mathbf{s_{c}}\mathbf{d_{t}}) \tag{12}\]  
where \(\mathbf{0}\leq\mathbf{s_{t}}\), \(\mathbf{s_{c}}\leq\mathbf{1}\), \(\mathbf{s_{t}}\), and \(\mathbf{s_{c}}\) are the relational expressions after stiffness recovery. \[\left\{ \begin{array}{l}\mathbf{s_{t}}=\mathbf{1}-\mathbf{\omega_{t}}\mathbf{r}\big{(} \widehat{\mathbf{\sigma}}\big{)},\mathbf{0}\leq\mathbf{\omega_{t}}\leq\mathbf{1}\\ \mathbf{s_{c}}=\mathbf{1}-\mathbf{\omega_{c}}\big{[}\mathbf{1}-\mathbf{r}\big{(}\widehat{\mathbf{\sigma }}\big{)}\big{]},\mathbf{0}\leq\mathbf{\omega_{c}}\leq\mathbf{1}\end{array} \right. \tag{13}\]  
where \(\mathbf{\omega_{t}}\) and \(\mathbf{\omega_{c}}\) are the weighting factors of stiffness recovery related to the material properties. Figure 1 shows the stiffness recovery curve of the concrete damage model when the weighting factors are \(\mathbf{\omega_{t}}=\mathbf{0}\) (compression \(\rightarrow\) tension) and \(\mathbf{\omega_{c}}=\mathbf{1}\) (tension \(\rightarrow\) compression) under uniaxial alternating loads. When the plastic damage model is used, the damage may cause the degradation of the concrete structure's stiffness. Based on it, the influence of concrete damage and cracking on the stress state of the structure can be simulated. At the same time, the concrete damage and cracking have a significant impact on the permeability characteristics of the structure. The material element is composed of a damaged phase and an undamaged phase. The element permeability coefficient is calculated as follows [37]:  
\[\mathbf{k}=(\mathbf{1}-\mathbf{d})\mathbf{k_{m}}+\mathbf{dk_{d}}\Big{(}\mathbf{1}+\varepsilon_{v}p^{ \prime\prime}\Big{)}^{3} \tag{14}\]  
where \(\mathbf{k_{m}}\) is the permeability coefficient of the undamaged phase, and \(\mathbf{k_{d}}\) is the permeability coefficient of the damaged phase. Assuming that no damage occurs in the case of elastic deformation, while plastic deformation and damage occur simultaneously, then, the plastic volumetric strain of the damaged phase is \(\varepsilon_{v}p^{\prime\prime}=\mathbf{dk_{v}^{p}}\), in which \(\varepsilon_{v}^{p}\) is the plastic volumetric strain of the element.