## 2 Methodology
### Concrete Damaged Plasticity Model (CDPM)
#### 2.1.1 Stress-Strain Relations in CDPM
The Macauley bracket \(\langle\cdot\rangle\) is defined by \(\langle x\rangle=\frac{1}{2}(|x|+x)\). If the eigenvalues of the plastic strain rate tensor \(\hat{\dot{\varepsilon}}_{i}\) (\(i=1,2,3\)) are ordered such that \(\overset{:pl}{\varepsilon}^{pl}_{\text{max}}=\hat{\dot{\varepsilon}}_{1}\geq \hat{\dot{\varepsilon}}_{2}\geq\hat{\dot{\varepsilon}}_{3}=\overset{:pl}{ \varepsilon}^{pl}_{\text{min}}\), the evolution equation for general multi-axial stress conditions can be expressed in the following matrix form:  
\[\overset{:pl}{\tilde{\varepsilon}}^{pl}= \begin{bmatrix}\overset{:pl}{ \tilde{\varepsilon}}_{t}^{pl}\\ \overset{:pl}{\tilde{\varepsilon}}_{c}^{pl}\end{bmatrix} =\mathbf{\hat{h}}( \overset{\overset{-}{\sigma}}{\sigma},\overset{\varepsilon}{\varepsilon }^{pl})\cdot\overset{:pl}{\tilde{\varepsilon}}^{pl},\mathbf{\hat{h}}( \overset{\overset{-}{\sigma}}{\sigma},\overset{\varepsilon}{\varepsilon}^{ pl})= \begin{bmatrix}\overset{\hat{\varepsilon}}{\varepsilon}_{1}\\ 0&0&-\bigg{(}1-r(\overset{\overset{-}{\sigma}}{\sigma})\bigg{)}\end{bmatrix} , \overset{:pl}{\tilde{\varepsilon}}^{pl}= \begin{bmatrix}\hat{\dot{\varepsilon}}_{1} \\ \hat{\dot{\varepsilon}}_{2}\\ \hat{\dot{\varepsilon}}_{3}\end{bmatrix} . \tag{7}\]The yield function \(F(\overset{-}{\mathbf{\sigma}},\overset{-}{\mathbf{\varepsilon}}^{pl})\) represents a surface in effective stress space, which determines the states of failure or damage. For the plastic-damage model,  
\[F(\overset{-}{\mathbf{\sigma}},\overset{-}{\mathbf{\varepsilon}}^{pl})\leq 0. \tag{8}\]  
Plastic flow is governed by a flow potential \(G\) according to the flow rule [9]:  
\[\dot{\mathbf{\varepsilon}}^{pl}=\dot{\lambda}\frac{\partial G(\overset{-}{\mathbf{ \sigma}})}{\partial\overset{-}{\mathbf{\sigma}}}, \tag{9}\]  
where \(\dot{\lambda}\) is the nonnegative plastic multiplier. The plastic potential is also defined in the effective stress space, and the CDPM uses non-associated plasticity. Within the context of the scalar-damage theory, the stiffness degradation is isotropic and characterized by a single degradation variable \(d\). The evolution of the degradation variable is governed by hardening variables \(\overset{-pl}{\mathbf{\varepsilon}}^{pl}\) and the effective stress \(\overset{-}{\mathbf{\sigma}}\); that is, \(d=d\overset{-}{(\overset{-}{\mathbf{\sigma}},\overset{-pl}{\mathbf{\varepsilon}}^{pl})}\). Similarly, by decomposing the degradation variable into tensile and compressive damage variables, [PERSON] assumes:  
\[1-d=(1-s_{\text{t}}d_{\text{c}})(1-s_{\text{c}}d_{\text{t}}), \tag{10}\]  
where \(s_{\text{t}}\) and \(s_{\text{c}}\) are functions of the stress state and are used to simulate stiffness recovery effects due to stress reversals. They are defined as follows [9]:  
\[s_{\text{t}}\overset{\text{def}}{\equiv}1-\omega_{\text{r}}r(\overset{\Delta} {\mathbf{\sigma}})\;(0\leq\omega_{\text{t}}\leq 1);s_{\text{c}}\overset{\text{def}}{ \equiv}1-\omega_{\text{c}}(1-r(\overset{\Delta}{\mathbf{\sigma}}))\;(0\leq\omega_{ \text{c}}\leq 1), \tag{11}\]  
where \(\omega_{\text{t}}\) and \(\omega_{\text{c}}\) are weight factors and control the recovery of the tensile and compressive stiffness upon load reversal, and the default values are \(\omega_{\text{t}}=0\) and \(\omega_{\text{c}}=1\) for concrete. In summary, the CDPM's elastic-plastic response is described in terms of the effective stress and the hardening variables by Equations (2), (4), (8) and (9), where \(\dot{\lambda}\) and \(F\) obey the Kuhn-Tucker conditions of \(\dot{\lambda}F=0;\dot{\lambda}\geq 0;F\leq 0\). The above constitutive relations for the elastic-plastic response are decoupled from the stiffness degradation response Equation (3), which makes the CDPM attractive for an effective numerical implementation.