## 2 Methodology
### Concrete Damaged Plasticity Model (CDPM)
#### 2.1.1 Stress-Strain Relations in CDPM  
The stress-strain relationship for general loading is given by the scalar damaged elasticity equation [9]:  
\[\mathbf{\sigma}=(1-d)\mathbf{D}_{0}^{d}:(\mathbf{\epsilon}-\mathbf{\varepsilon}^{pl})= \mathbf{D}^{el}:(\mathbf{\epsilon}-\mathbf{\varepsilon}^{pl}), \tag{1}\]  
where \(\mathbf{\sigma}\) and \(\mathbf{\varepsilon}\) are the stress and strain tensors, respectively; \(\mathbf{D}_{0}^{el}\) is the initial (undamaged) elastic stiffness tensor; \(\mathbf{D}^{el}\) is the degraded elastic stiffness tensor; \(\mathbf{\varepsilon}^{pl}\) is the plastic strain tensor. \(d\) is the scalar stiffness degradation variable, which can take values in the range from zero (undamaged material) to one (fully damaged material). Damage associated with the failure mechanisms of the concrete (cracking and crushing), therefore, results in a reduction in the elastic stiffness. Following the usual notions of continuum damage mechanics, the effective stress is defined as [9]:  
\[\overset{-}{\sigma}\overset{\text{def}}{=}\mathbf{D}_{0}^{el}:(\varepsilon- \varepsilon^{pl}). \tag{2}\]  
The Cauchy stress is related to the effective stress through the scalar degradation relation:  
\[\sigma=(1-d)\overset{-}{\sigma}. \tag{3}\]  
For any given cross-section of the material, the factor \((1-d)\) represents the ratio of the effective load-carrying area (i.e., the overall area minus the damaged area) to the overall section area. In the absence of damage, the effective stress is equivalent to the Cauchy stress. When damage occurs, however, the effective stress is more representative than the Cauchy stress because the effective area resists the external loads. It is, therefore, convenient to formulate the plasticity problem in terms of the effective stress \(\sigma\) and hardening variables \(\varepsilon^{pl}\). Damaged states in tension and compression are characterized independently by two hardening variables, \(\tilde{\varepsilon}_{t}^{pl}\) and \(\tilde{\varepsilon}_{c}^{pl}\), which are referred to as equivalent plastic strains in tension and compression, respectively. In general, the evolution of the hardening variables in rate format is given by an expression of the form [9]:  
\[\overset{-}{\varepsilon}^{pl}=\mathbf{h}(\overset{-}{\sigma},\overset{-}{ \varepsilon}^{pl})\cdot\dot{\varepsilon}^{pl}. \tag{4}\]  
Please note that an additive strain rate decomposition is assumed for the rate-independent CDPM as:  
\[\dot{\varepsilon}=\dot{\varepsilon}^{el}+\dot{\varepsilon}^{pl}, \tag{5}\]  
where \(\dot{\varepsilon}\) is the total strain rate, \(\dot{\varepsilon}^{el}\) is the elastic part of the strain rate, and \(\dot{\varepsilon}^{pl}\) is the plastic part of the strain rate. Microracking and crushing in the concrete are represented by increasing values of the hardening variables. These variables control the evolution of the yield surface (or failure surface) and the degradation of the elastic stiffness. Abaqus assumes that the equivalent plastic strain rates are evaluated according to the expressions [9]:  
\[\overset{:pl}{\tilde{\varepsilon}}_{t}^{pl}\overset{\text{def}}{=}r( \overset{\overset{-}{\sigma}}{\sigma})\overset{\dot{\varepsilon}}{\varepsilon }^{pl}_{\text{max}};\overset{:pl}{\tilde{\varepsilon}}_{c}^{pl}\overset{ \text{def}}{=}-\bigg{(}1-r(\overset{\overset{-}{\sigma}}{\sigma})\bigg{)} \overset{\dot{\varepsilon}}{\varepsilon}^{pl}_{\text{min}}, \tag{6}\]  
where \(\overset{:pl}{\varepsilon}^{pl}_{\text{max}}\) and \(\overset{:pl}{\varepsilon}^{pl}_{\text{min}}\) are the maximum and minimum eigenvalues of the plastic strain rate tensor, respectively. \(r(\overset{\overset{-}{\sigma}}{\sigma})\overset{\text{def}}{=}\frac{\sum_ {i=1}^{3}(\hat{\varphi}_{i})}{\sum_{i=1}^{3}|\hat{\varphi}_{i}|},0\leq r( \overset{\overset{-}{\sigma}}{\sigma})\leq 1\) is a stress weight factor that is equal to one if all principal effective stresses \(\hat{\mathcal{H}}_{i}\) (\(i=1,2,3\)) are positive and equal to zero if they are negative.