## 5 Data Processing and Definition of the EAR Index
### Definition of the Performance Coefficients
#### 5.2.2 Fatigue Coefficient  
One of the most widely used parameters in the literature for the mechanical characterization of bituminous materials is represented by the complex stiffness modulus \(E^{*}\). This parameter depends on temperature, loading speed, age, and other characteristics of the mixture. As suggested by the American standard NCHRP Project 1-37A Report [35] at level 3 of the analysis, it is possible to evaluate \(E^{*}\) by means of the sigmoidal function at the reference temperature of 70 \({}^{\circ}\)F (i.e., about 21.1 \({}^{\circ}\)C):  
\[\log E^{*}=3.750063+0.02932\rho_{200}-0.001767\rho_{200}^{2}-0.00 2841\rho_{4}-0.0058097V_{A}+\\ -0.802208\frac{V_{beff}}{V_{beff+V_{A}}}+\frac{3.871977-0.0021 \rho_{4}+0.003958\rho_{38}-0.000017\rho_{38}^{2}+0.005470\rho_{34}}{1+e^{(-0.60 3313-0.313351\log f-0.393532\log\eta)}} \tag{12}\]  
where \(\eta\) is the viscosity of bitumen (equal to \(10^{6}\) Poise), \(f\) is the frequency of the load expressed in hertz, \(V_{A}\) is the percentage of void content in the asphalt mixture, \(V_{b,eff}\) is the percentage effective content of bitumen, \(\rho_{34}\), \(\rho_{38}\), \(\rho_{4}\) are the percentage retained on the \({}^{3}\!/_{4}\), \({}^{3}\!/_{8}\) and 4 sieve, respectively, \(\rho_{200}\) is the percentage passing on the 200 sieve.  
All these parameters are requested during the pre-qualification stage except the viscosity. However, this parameter can be derived, according to (13) on the base of the penetration grade of bitumen which is information available downstream the pre-qualification stage.  
\[\log\eta=10.5012-2.2601\log P_{\text{en}}+0.00389\log P_{\text{en}}^{2} \tag{13}\]  
\(P_{\text{en}}\) is the value of the penetration test of the asphalt binder. Once the complex stiffness modulus is derived, it is possible to use its absolute value \(E\) to evaluate the residual number of standard fatigue axles \(N_{\text{res,sa}}\) for each mixture through the following relationship:  
\[N_{\text{res,sa}}=k_{1}\epsilon^{-k_{2}}E^{-k_{3}} \tag{14}\]  
where \(\epsilon\) is the maximum horizontal strain at the base of the asphalt mixture. Three different reference values of \(\epsilon\) have been assumed in order to take into account the different typologies of pavement:  
* \(\epsilon=3.61\cdot 10^{-5}\) for dense graded asphalt concrete;
* \(\epsilon=3.49\cdot 10^{-5}\) for binder asphalt concrete;
* \(\epsilon=3.29\cdot 10^{-5}\) for base asphalt concrete. \(k_{1},k_{2}\) and \(k_{3}\) have been assumed as follows:
* \(k_{1}=0.0796\);
* \(k_{2}=3.291\);
* \(k_{1}=0.854\).  
It may be noted that, using (14), a value of \(N_{\text{res,sa}}\) referring to a standard 80 kN axles is obtained. This value has been reported to a value referred to standard axles of 120 kN (i.e., the reference load in the Italian context) through (15) [31, 32, 33, 34]:  
\[N_{\text{sa,120 kN}}=N_{\text{sa,80 kN}}\bigg{(}\frac{80}{120}\bigg{)}^{2} \tag{15}\]  
Finally, the coefficient to be applied at the EAR index can be evaluated, for each mixture, according to the equation:  
\[I_{2}=\frac{N_{\text{res,sa}}}{N_{\text{ref,sa}}} \tag{16}\]  
in which the number of cycles to fatigue failure, expressed in terms of 120 kN standard axle, is normalized by a factor \(N_{\text{ref,sa}}\) assumed equal to:  
* Eighty-four MESAL\({}_{120}\) for dense graded asphalt concrete;
* Ninety-one MESAL\({}_{120}\) for binder asphalt concrete;
* One-hundred-and-five MESAL\({}_{120}\) for base asphalt concrete.  
Furthermore, in this case, it is recalled that the coefficient \(I_{2}\) applies to the closed graded mixtures but it is set equal to one for all other kinds of mixtures.