## 2 Theoretical Background
### Damage-Coupled Viscoelastic Model  
[PERSON] [35] proposed the concept of continuum damage mechanics to describe the progressive decay of materials, where [PERSON] [36] introduced the continuity \(\zeta\) to define the reduction of area due to micro-damages in Equation (7):  
\[\varnothing=1-\zeta=\frac{A-\overline{A}}{A} \tag{7}\]  
where \(A\) is the damaged (apparent) area and \(\overline{A}\) is the real area (intact or undamaged area) carrying the load. In other words, \(\overline{A}\) is the resulted effective area after micro-damages (micro-cracks and micro-voids) are removed from the damaged area \(A\). \(\varnothing\) is the so-called damage density or damage variable. For cases of the isotropic (scalar) damage, the effective stress is given by Equation (8):  
\[\overline{\sigma}_{ij}^{t}=\frac{\sigma_{ij}^{t}}{1-\varnothing(t)}=\frac{S_{ij}^ {t}+\frac{1}{3}\sigma_{kk}^{t}\delta_{ij}}{1-\varnothing(t)} \tag{8}\]  
where \(\overline{\sigma}_{ij}^{t}\) is the effective stress tensor in the effective (undamaged) configuration. Generally, the damage density can be a function of stress, strain, temperature, and damage history. However, a damage law considering various influencing factors is not appropriate for predicting the damage growth in asphalt mixture since it is very difficult to express these influences quantitatively in one equation due to their mutual influence. [PERSON] [36] proposed a damage evolution law in terms of strain under exponent format, where he eliminated other influence factors from the evolution law. [PERSON] et al. [37] proposed an approximation for the damage variable relying that the strain is the most important factor. [PERSON] et al. [27] proposed a unified damage-couple viscoelastic model by using the Schapery model, where the damage variables are functions of maximum strain. [PERSON] et al. [32] proposed the first approximation of the damage evolution law as an exponential form of the total effective strain. As for asphalt binder damage evolution law, the Linear Amplitude Sweep (LAS) test is proposed by [PERSON] to specify the fatigue resistance [38; 39]. The LAS test is a cyclic sinusoidal torsion test that uses increasing loading amplitude to accelerate damage. The damage accumulation (\(D\)) in the asphalt binder sample can be calculated using in Equation (9) to in Equation (12) [40]:  
\[D(t)\cong\sum_{i=1}^{N}\left[\pi\gamma_{0}^{2}(C_{i-1}-C_{i})\right]^{\frac{ \alpha}{1+\alpha}}(t_{i-1}-t_{i})^{\frac{1}{1+\alpha}} \tag{9}\]  
\[G\prime(\omega)\cong|G^{*}|(\omega)\times\cos\delta(\omega) \tag{10}\]  
\[log\ G\prime(\omega)=m(log\ \omega)+b \tag{11}\]  
\[\alpha=1/m \tag{12}\]  
where \(C_{i}\cong|G*|(i)|/|G*|_{initial}\) which is the complex shear modulus for a given data point divided by its initial undamaged value and \(C_{i-1}\) is the complex shear modulus divided by its initial undamaged value at a former data point. \(\gamma_{0}\) is the applied shear strain amplitude at each loading cycle and \(\alpha\) is the exponent determining energy release rate. \(|G^{*}|(\omega)\) and \(\delta(\omega)\) are the dynamic modulus and phase angle at frequency \(\omega\), respectively. m is the slope of a best-fit straight line plotting with log \(\omega\) on the horizontal axis and log \(G\prime(\omega)\) on the vertical axis. Many numerical attempts have been made to investigate the damage behavior of asphalt mixture under certain conditions. However in these numerical models, the asphalt mixture is considered as being a homogenous material with equivalent properties. Thus, the damage density is treated as a constant at each Gaussian integral points for each incremental step. Asphal mixture is a particulate composite material consisting of aggregate, mastic, and air voids. The mechanical behavior of asphalt mixture strongly depends on the particular components. During the damage evolution process in heterogeneous materials, the amount of damage in each region of the material is related to the current strain.