## 2 Compaction Theories and Compactability in Asphalt Mixtures  
A compaction theory with a mathematical equation was firstly initiated to obtain the measurable parameters characterizing the compaction process of asphalt mixtures. Compaction resistance was introduced into the equation with mathematical terms [46]. The physical assumption is that the increase in density d\(\rho\), which is achieved by the compaction work d\(S\), is proportional to the compaction potential. The compaction is the difference between the current density and a theoretical maximum density that is achieved after infinite compaction work. If the proportional constant is set to \(1/C\), the following equation can be found:  
\[\frac{\mathrm{d}\rho}{\mathrm{d}S}=\frac{1}{C}(\rho_{\infty}-\rho_{0}) \tag{1}\]  
By solving Equation (1) using the initial condition \(\rho_{0}=\rho(S=0)\), an exponential equation describing the compaction process is obtained:  
\[\rho(S)=\rho_{\infty}-(\rho_{\infty}-\rho_{0})\cdot e^{\frac{S}{C}} \tag{2}\]  
where \(\rho(S)=\) current density as a function of compaction work (kg/m\({}^{3}\)), \(S=\) compaction work (Nm), \(\rho_{\infty}\) = calculated maximum density (\(\rho(S\)\(\rightarrow\)\(\infty\))), \(\rho_{0}\) = initial density (\(\rho(S=0)\)), and \(C=\) compaction resistance (Nm). It assumes constant temperature in the material during the entire compaction process. The compaction process for Equation (2) is presented in Figure 1. The influence of temperature on the coefficient of the exponential compaction equation was introduced in two principally different ways. In the first case, if the mix is easy to compact, the same maximum compaction level (\(\rho_{\infty}\)) is reached even at a lower temperature, but the compaction resistance will increase considerably. In the second case, if the mix is very difficult to compact, the compaction resistance remains constant; instead, the final compaction level decreases as the temperature decreases. Consequently, the compaction of typical mixes results in both decreasing the maximum density and increasing the compaction resistance as the temperature decreases. In order to estimate the compaction resistance at a temperature that is not examined experimentally, a mathematical relationship between the compaction temperature and the compaction resistance was introduced with a reference temperature of 135 \({}^{\circ}\)C. However, to estimate the effects on temperature is generally ambiguous when using a modified asphalt binder. Based on the expected similarity between the asphalt mixture and the granular material, the compaction of the asphalt mixture was characterized by means of the parameters \(\tau\) and \(\phi\) (according to the Mohr-Coulomb theory for granular materials). In addition, the parameter \(\eta\) to capture the viscous component of the asphalt mixture was introduced. It was assumed that the resistance against deformation of the asphalt mixture could be represented by three physical quantities such as the angle of internal friction (\(\phi\)), the initial resistance (\(\tau\)), and the viscosity of the mass (\(\eta\)), which denotes the influence of the viscosity on the shear resistance of the asphalt mixtures. This latter parameter takesinto account that the position of the Mohr-Coulomb failure line depends on the viscosity of the mass, and thus on the temperature of the material or loading time (Figure 2). Later on, [PERSON] et al. [29] assessed the effects of several mixture characteristics on their compaction parameters to determine the asphalt mixture temperature range in which the desired density can be achieved using the Mohr-Coulomb theory. However, it is also difficult to explain the compaction process in that the asphalt binder stiffness can be changed at different loading and temperature conditions. Thus, the compaction theories above are quite simple, but have some difficulty in explaining the material properties and compaction conditions.