## 3 Materials and Methods
### MSCR Data Modelling  
To better understand rutting potential at binder-, mastic-, and fine-mortar scales, a further elaboration of high-temperature experimental data collected through MSCR tests was then carried out. This allowed achievement of a more detailed description of the visco-elastic deformations exhibited by blends during each test cycle, thus differentiating eventual different behaviors of materials finally characterized by similar creep-recovery summary results (i.e., strain recoveries and compliances). To accomplish this objective, each cycle was modelled using an analytical approach able to model the whole creep-recovery curve of tested materials at each loading cycle [34]. The use and validation of such a model, originally validated for polymer- and rubber-modified asphalt binders at the mastics- and fine-mortar scale, for testing unconventional materials such as the steel-slag fine aggregates used for the solar collector asphalt layers is rather innovative.  
Based on the abovementioned modelling approach, fractional calculus was used to fit the experimental results based on the rheological model schematized in Figure 3. In analogy with the work of [PERSON] et al. [34], a springpot and dashpot in series were considered to model the visco-elastic measured response. The first element (springpot) is obtained, replacing the first-order derivatives of generalized rheological models (e.g., Maxwell, Kelvin-Voigt models) with derivatives of order \(\alpha\)[35]. Its constitutive relation is represented in Equation (4), where \(\alpha\) ranges from 0 and 1. For \(\alpha=0\), the element turns in a spring, whereas a conventional dashpot is considered when \(\alpha=1\). Based on this hypothesis, parameter \(b_{1}\) assumes a different physical meaning depending on \(\alpha\) value (it stands for the element stiffness when \(\alpha=0\) or the viscosity when \(\alpha=1\)). The second element (dashpot) was introduced to attain a residual permanent strain at the end of each test cycle (\(b_{2}\) indicated its viscosity).  
\[\sigma(t) =\ b_{1}\ \frac{d^{\alpha}\varepsilon(t)}{dt^{\alpha}} \tag{4}\]  
Under the proposed assumptions, the overall constitutive law of the model is given in Equation (5):  
\[\varepsilon(t)=\bigg{\{}\bigg{[}\frac{t^{\alpha}}{b_{1}\Gamma(1+\alpha)}+ \frac{t}{b_{2}}\bigg{]}U(t)-\bigg{[}\frac{(t-t^{*})^{\alpha}}{b_{1}\Gamma(1+ \alpha)}+\frac{(t\ -\ t^{*})}{b_{2}}\bigg{]}U(t\ -\ t^{*}\bigg{)}\bigg{\}}\sigma_{0} \tag{5}\]  
where, \(\alpha\), \(b_{1}\), and \(b_{2}\) are the already described parameters; \(\sigma_{0}\) indicates the stress level applied to cycle, and \(t^{*}\) expresses the time at which unloading starts (1 s); \(\Gamma\) is the Euler-Gamma function; \(U\) is the unit step Heaviside function. Adopting such a mathematical approach, the use of complex power-laws commonly associated to the creep-loading  
Figure 3: Creep-recovery proposed model: springpot in series with dashpot.  
loading configuration of traditional rheological models was prevented, involving the fractional model with non-integer order derivatives. Additional details about the proposed model can be found elsewhere [34; 35; 36].