# Induction Heating-Assisted Compaction in Porous Asphalt Pavements: A Computational Study
## 3 Simulation of Induction Heating-Assisted Compaction in Porous Asphalt
### Modeling of Asphalt Mix
#### 3.2.3 Model Parameters Determination  
The previously mentioned five parameters had a great influence on the degree of asphalt compaction and were determined by the mix temperature. However, the compactability of the mix, which was influenced by the temperature as well, was determined by the air void content. Research [25] has shown that the air void content is linearly related to the coordination number The coordination number of a particle is defined as the total number of particles which are in contact with it. In this paper, the air void content of the mix was changed by compressing the same pavement DEM model, and the relationship was obtained as shown in Figure 5:  
\[n=n_{0}+A\overline{n}_{c}, \tag{1}\]  
where \(n\) is the air void content, \(\overline{n}_{c}\) is the average coordination number of aggregates, and \(n_{0}\) and \(A\) are coefficients. The relationship of the dynamic module of asphalt mortar under different temperatures was investigated by [PERSON] et al. [26], and it was expressed as shown in Equation (2),  
\[E^{*}=E_{0}e^{-\alpha T}, \tag{2}\]  
where \(E^{*}\) is the dynamic modulus of the mortar (MPa), \(T\) is the temperature, and \(E_{0}\) and \(\alpha\) are the complex parameters. In this bond model, the stiffness of asphalt mortar was only produced by a cluster of springs. The dynamic modulus was replaced by the stiffness, and the relation is given in Equation (3),  
\[K=\frac{E^{*}}{L}, \tag{3}\]  
where \(K\) is the stiffness of the bond (MPa/m) and \(L\) is the distance between the centroid of two particles that are bonded. This relationship shows that the value of stiffness is reduced when two particles are moving away and by keeping dynamic modulus constant. In this research, the dynamic modulus of LTPA mortar was assumed to obey the curve shown in Figure 6, and the stiffness in normal direction was kept the same as that in shear direction. Figure 5: Determination of model parameters: The relationship between air void content and the average coordination number. Since the asphalt mortar was assumed to be a viscoelastic material, a peak value of stress and permitted strain existed. When a bond exceeds any one of these limits, the bond can be treated as broken and the rheological property of mortar is easily reflected. The tensile and shear strength of asphalt mortar have been investigated and the obtained results demonstrate that the strength is exponentially related to temperature [27; 28; 29; 30] (Figure 7 and Equation (4)):  
\[\sigma_{c}=\sigma_{0}e^{-\beta T}, \tag{4}\]  
where \(\sigma_{c}\) is the tensile strength and \(\sigma_{0}\) and \(\beta\) are the corresponding parameters. The elastic energy stored in the springs of the bond could not be released, as it was obviously not consistent with the purpose of simulation. Therefore, a critical strain should be applied onto the model. Then the effective strength of mortar can be written as  
\[\overline{\sigma}_{c}=min(\sigma_{c},\,\overline{\varepsilon}_{r}E^{*}), \tag{5}\]  
where \(\overline{\sigma}_{c}\) is the effective strength and \(\overline{\varepsilon}_{r}\) is the critical strain. In this research, \(\overline{\varepsilon}_{r}\) is given by Equation (6),  
\[\overline{\varepsilon}_{r}=0.017e^{-0.018T}. \tag{6}\]  
Figure 6: Determination of model parameters: Dynamic modulus of low temperature porous asphalt (LTPA) mortar. Figure 7: Determination of model parameters: Strength curves of asphalt mortar from the literature. Since the failure mode of asphalt mortar was controlled by the Mohr-Coulomb strength criterion, the shear strength of the mortar was determined by the normal stress \(\sigma\), the cohesion \(c\), and the internal friction angle \(\varphi\)[31]. Herein, only the angle of internal friction is discussed, which was influenced by the temperature and the compactability of the mix [29; 30]. This parameter referred not only to the attribution of asphalt mortar but also to the whole interaction of contacted particles. In [PERSON] et al.