## 3 Damage-Healing Formulations
### Micro-Damage Healing Models for Asphalt Mixtures  
In [115], an elastic-viscoelastic model with healing for asphalt concrete subjected to fatigue loading was proposed. The model was extended from the work presented in [116] in which the pseudo-strain variables are given in [117] and adopted to eliminate the dependency of the stress-strain material behavior to time. The irreversible thermodynamic framework was used to simulate the healing of micro-damage. Growing damage was simulated using uniaxial viscoelastic constitutive equations that are extended to account for the micro-damage healing. Uniaxial tensile tests under cyclic loading were conducted under controlled-strain and controlled-stress models with rest periods. In order to induce damage in the specimens, two stress-strain levels were used in the tests. The rest periods introduced during each test vary from 0.5 to 32 min. The variation of the material stiffness before and after the rest period was studied as a function of the number of cycles (Figure 5). Region I in Figure 5 depicts the reduction of the stiffness due to the damage evolution without a rest period, while region II depicts the reduction of the stiffness due to the damage evolution after rest period. After the introduction of the rest period, it was shown that the stiffness increases from point B to point A due to micro-damage-healing and decreases after damage of the healed material. Based on the experimental results and stiffness variation in regions I and II, the following healing function was proposed:  
\[H=\left[S_{B,j}^{R}+C_{2}\left(S_{2,j}\right)\right]C_{3}\left(S_{3,i}\right) -C_{1}\left(S_{1n}\right)-\sum_{j=1}^{i-1}\left(S_{B,j}^{R}-S_{C,j}^{R}\right), \tag{27}\]  
\[\text{when }S^{R}>S_{B,i}^{R}\text{ (region I),}\]  
\[H=\sum_{j=1}^{i}\left(S_{B,j}^{R}-S_{C,j}^{R}\right), \tag{28}\]  
\[\text{when }S^{R}<S_{B,i}^{R},\]  
where \(\left(S_{2,i}\right)\) represents the healing evolution during the ith rest period and \(\left(S_{3,i}\right)\) represents the damage evolution after the ith rest period. \(C_{1}(S_{1n})\) and \(\left(S_{1n}\right)\) are the material function and the normalized damage variable. Several experimental investigations were carried out for the study of the micro-damage healing of asphalt mixtures (e.g., [118; 119; 120; 121; 122; 123; 124]). Although the implemented micro-damage-healing model was able to describe the hysteretic behavior under controlled-strain and controlled-stress modes, the identification of the experimental data to simulate the healing behavior is not an easy task. Another micro-damage-healing model applied to asphalt mixtures subjected to fatigue loads was proposed in [125]. The damage healing model was extended from the viscoelastic, viscoplastic and viscodamage model. The authors defined a healing variable which is a function of the healing time and history, damage level and temperature. The proposed healing variable takes the following expression:  
\[h=\Gamma^{h}\left(T\right)\left(1-\bar{\phi}\right)^{b_{1}}\left(1-h\right)^{ b_{2}}, \tag{29}\]  
where \(h\) is the rate of the healing variable, \(\Gamma^{h}\) is the healing viscosity parameter and \(b_{1}\) and \(b_{2}\) are material constants. \(\Gamma^{h}(T)\) is a function of temperature and is the parameter that determines the speed of the healing. It is expressed as  
\[7\Gamma^{h}(T)=\Gamma_{0}^{h}exp[-\delta_{3}(1-\frac{T}{T_{0}})], \tag{30}\]where \(\Gamma_{0}^{h}\) and \(\delta_{3}\) are the healing viscosity parameter at temperature \(T_{0}\) and the healing-temperature coupling parameter, respectively. According to Equation (29), it was assumed that the healing starts to evolve once the temperature reaches a certain reference level (temperature threshold) and decreases when the temperature is less than the reference level. The model was applied to predict the behavior of creep-recovery tests in compression and in tension.