## 2 Methodology
### Concrete Damaged Plasticity Model (CDPM)
#### 2.1.3 Input of Uniaxial Behavior  
The scalar degradation variable \(d\) is reduced to \(d=d_{\text{t}}\) for uniaxial tension and \(d=d_{\text{c}}\) for uniaxial compression. The stress-strain relationship response of concrete subjected to uniaxial tension or compression is shown in Figure 1, where the tension part is enlarged for a clearer view. The tensile and compressive stresses in concrete in Figure 1 can be expressed as [9]  
\[\sigma_{\text{t}}=(1-d_{\text{t}})E_{0}(\varepsilon_{\text{t}}-\tilde{ \varepsilon}_{\text{t}}^{\text{pl}});\;\sigma_{\text{c}}=(1-d_{\text{c}})E_{0} (\varepsilon_{\text{c}}-\tilde{\varepsilon}_{\text{c}}^{\text{pl}}), \tag{15}\]  
and the effective uniaxial cohesion stresses, \(\overline{\sigma}_{\text{t}}\) and \(\overline{\sigma}_{\text{c}}\), are given as:  
\[\overline{\sigma}_{\text{t}}=\frac{\sigma_{\text{t}}}{(1-d_{\text{t}})}=E_{0} (\varepsilon_{\text{t}}-\tilde{\varepsilon}_{\text{t}}^{\text{pl}});\; \overline{\sigma}_{\text{c}}=\frac{\sigma_{\text{c}}}{(1-d_{\text{c}})}=E_{ 0}(\varepsilon_{\text{c}}-\tilde{\varepsilon}_{\text{c}}^{\text{pl}}). \tag{16}\]  
\(d_{\text{t}}\) and \(d_{\text{c}}\) are damage variables of concrete under tension and compression, respectively, which could be derived as \(d_{\text{t}}=1-\sqrt{\sigma_{\text{t}}/(E_{0}\varepsilon_{\text{t}})}\) and \(d_{\text{c}}=1-\sqrt{\sigma_{\text{c}}/(E_{0}\varepsilon_{\text{c}})}\) based on the Sidoroff energy equivalence principle [29], \(E_{0}\) is concrete's initial modulus of elasticity; \(\varepsilon_{\text{t}}\) and \(\varepsilon_{\text{c}}\) are the tensile and compressive strains of concrete, respectively; \(\tilde{\varepsilon}_{\text{t}}^{\text{pl}}\) and \(\tilde{\varepsilon}_{\text{c}}^{\text{pl}}\) are the equivalent plastic strains of concrete. The equivalent plastic strains are defined as a function of cracking strain \(\tilde{\varepsilon}_{\text{t}}^{\text{ck}}\) and inelastic strain \(\tilde{\varepsilon}_{\text{c}}^{\text{in}}\), as shown in Equation (17):  
\[\tilde{\varepsilon}_{\text{t}}^{\text{pl}}=\tilde{\varepsilon}_{\text{t}}^{ \text{ck}}-\frac{d_{\text{t}}}{1-d_{\text{t}}}\frac{\sigma_{\text{t}}}{E_{0} };\;\tilde{\varepsilon}_{\text{c}}^{\text{pl}}=\tilde{\varepsilon}_{\text{c}}^ {\text{in}}-\frac{d_{\text{c}}}{1-d_{\text{c}}}\frac{\sigma_{\text{c}}}{E_{0}}. \tag{17}\]  
Figure 1: Sketch of monotonic behavior of CDPM under uniaxial tension and compression. Besides plastic deformation, the model explicitly considers a loss in concrete’s modulus of elasticity due to damage caused by compressive crushing and tensile cracking. The cracking strain \(\tilde{\epsilon}_{\rm t}^{\rm k}\) and inelastic strain \(\tilde{\epsilon}_{\rm c}^{\rm in}\) are defined as the total strain minus the elastic strain in the undamaged state and can be expressed as:  
\[\tilde{\epsilon}_{\rm t}^{\rm ck}=\epsilon_{\rm t}-\frac{\sigma_{\rm t}}{E_{0}}; \,\tilde{\epsilon}_{\rm c}^{\rm in}=\epsilon_{\rm c}-\frac{\sigma_{\rm c}}{E_{0}}. \tag{18}\]  
Equations (15)-(18) determine the input of uniaxial mechanical behavior parameters. Data pairs of (\(\sigma_{\rm t}\tilde{\epsilon}_{\rm c}^{\rm ck}\)) and (\(d_{\rm t}\tilde{\epsilon}_{\rm c}^{\rm ck}\)) define CDPM's tension stiffening and damage; data pairs of (\(\sigma_{\rm c}\tilde{\epsilon}_{\rm c}^{\rm in}\)) and (\(d_{\rm c}\tilde{\epsilon}_{\rm c}^{\rm in}\)) define stress hardening, followed by strain softening and damage in compression. The constitutive theory in this section can capture the effects of irreversible damage associated with the failure mechanisms that occur in concrete as long as the confining pressures are less than four or five times the ultimate compressive strength in uniaxial compressive loading. Although CDPM can characterize the nonlinear mechanical behavior of concrete under cyclic or dynamic loading, the computational cost associated with cycle-by-cycle simulation in high-cycle fatigue is enormously high [10].