## 2 Material Constitutive Model and Interface Cohesive Zone Model  
### Static and Fatigue Damage Constitutive Model of Concrete  
Among a number of damage models [25; 27], a fibre bundle-plastic chain model (BCM) [25] is developed based on the classical fibre bundle model (FBM) for characterising the mechanical behaviours of quasi-brittle materials under external loading. The damage model BCM was verified to be able to capture the stochastic microstructure and the crack/damage behaviours of concrete material and describe the stress-strain curve under fatigue loading precisely and conveniently [25]. According to the constitutive law recommended in the concrete structure design specification [27], the concrete static constitutive model can be determined by the regression method [25]:  
\[\sigma^{\pm}=(1-d^{\pm})E_{0}\varepsilon^{\pm} \tag{1}\]The damage evolution law of tensile concrete is as follows. \[d^{+}=A_{1}^{+}+\left(A_{2}^{+}-A_{1}^{+}\right)\left[\frac{c}{1+10^{p_{1}\cdot \left(x_{1}-x\right)}}+\frac{1-c}{1+10^{p_{2}\cdot\left(x_{2}-x\right)}}\right] \tag{2}\]  
The damage evolution law of compressive concrete is as follows. \[d^{-}=\frac{A_{1}^{-}-A_{2}^{-}}{1+\left(x/x_{0}\right)^{p_{3}}}+A_{2}^{-} \tag{3}\]  
\[x=\frac{\varepsilon}{\varepsilon_{\mathrm{k}}^{\pm}} \tag{4}\]  
where \(\sigma^{\pm}\) and \(\varepsilon^{\pm}\) denote the concrete tensile (compressive) stress and strain, respectively; \(E_{0}\) is the initial elastic modulus of concrete; \(A_{1}{}^{+}\), \(A_{2}{}^{+}\), \(c\), \(p_{1}\), \(p_{2}\), \(x_{1}\), and \(x_{2}\) are parameters related to the tensile damage of concrete material and can be obtained from the literature by the authors [25]; \(A_{1}{}^{-}\), \(A_{2}{}^{-}\), \(p_{3}\), and \(x_{0}\) are parameters related to the compressive damage of concrete material that can be obtained from the literature by the authors [25]; and \(\varepsilon^{\pm}{}_{\mathrm{k}}\) is the uniaxial tensile (compressive) strain corresponding to the peak strain of the concrete. Furthermore, the fatigue degradation of concrete under cyclic loading is characterised by elastic deformation damage and irreversible deformation damage [25]. Considering the time-dependent and random patterns of the concrete material damage, the concrete fatigue constitutive law is established by the regression method [25]:  
\[\sigma_{n}^{\pm}=(1-d_{n}^{\pm})E_{0}\varepsilon_{n}^{\pm} \tag{5}\]  
The concrete fatigue damage variable evolution law is  
\[d_{n}^{\pm}=d_{1}^{\pm}+d_{0}^{\pm}\cdot\left(d_{N}^{\pm}-d_{1}^{\pm}\right) \tag{6}\]  
where the normalised damage variable is  
\[d_{0}^{\pm}=A_{3}^{\pm}\left(-\frac{n/N}{n/N-A_{4}^{\pm}}\right)^{\frac{1}{ \sigma_{5}^{\pm}}} \tag{7}\]  
For compressive concrete, the irreversible deformation damage variable is  
\[d_{\mathrm{i}n}^{-}=\left\{ \begin{array}{ll}d_{n}^{-},&d_{n}^{-}\leq 0.2\\ \frac{1}{-5d_{n}^{+}+6},&d_{n}^{-}>0.2\end{array} \right. \tag{8}\]  
while the irreversible deformation damage variable of tensile concrete is  
\[d_{\mathrm{i},n}^{+}=\frac{d_{n}^{+}-d_{\mathrm{e},n}^{+}}{1-d_{\mathrm{e},n} ^{+}} \tag{9}\]  
where the fatigue elastic deformation damage variable is  
\[d_{\mathrm{e},n}^{+}=\left\{ \begin{array}{ll}0,&\varepsilon^{+}\leq 115.5 \times 10^{-6}\\ 1-\left(\frac{\varepsilon^{+}}{115.5\times 10^{-6}}\right)^{-1.05},&\varepsilon^{+}>11 5.5\times 10^{-6}\end{array} \right. \tag{10}\]  
The fatigue life can be calculated by the following formula [28].