# From Complex Modulus E* to Creep Compliance D(t): Experimental and Modeling Study
## 3 Results and Discussion
### Direct Tensile and Direct Compression Creep Tests Results  
Creep tests were performed using three configurations namely direct tensile test, direct compression test and indirect tensile test. As shown in Figure 10, for direct tensile tests and direct compression tests, a static stress was applied to the cylindrical samples. This configuration allows to have a homogeneous stress distribution, so the creep compliance can be calculated directly using the Equation (17).  
Figure 10: Creep loading for direct tensile and direct compression tests.  
Figure 9: The error between D(t)GKV model and D(t) from 2S2P1D coupled to Kopelman approximation.  
The load was applied in 0.1 s. The amplitude of force and stress applied are presented in Table 1. For the first 10 to 15 s, the applied load was not stable. Creep compliance was calculated (Equation (17)) when the load is stabilized at \(\pm\) 2% of the target value at t equal to 0.1, 1, 3, 10, 30, 100 and 300 s.  
\[D(t)=\frac{\varepsilon(t)}{\sigma_{max}} \tag{17}\]  
The results of creep compliance are presented in Figure 11 at different test temperatures (isotherms at \(-20\)\({}^{\circ}\)C, \(-10\)\({}^{\circ}\)C, \(0\)\({}^{\circ}\)C and \(10\)\({}^{\circ}\)C). According to the results, we notice that the creep compliances from direct tensile and from direct compression are the same. The majority of points are superposed on the equality line with an error interval between \(\pm\)3% and a maximum error of 18% at high creep compliance value.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Temperature (\({}^{\circ}\)C)** & **Force Max (N)** & **Stress Max (MPA)** \\ \hline \(-20\) & 1800 & 0.41 \\ \(-10\) & 1500 & 0.34 \\ \(0\) & 1200 & 0.27 \\ \(10\) & 500 & 0.11 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Forces and stresses amplitude applied for direct tensile and direct compression tests.  
Figure 11: _Cont._  
This results are contradictory with the results presented by NCHRP [10]. The explanation could be that the results presented by NCHRP were out of the VEL domain. As mentioned before, to verify that our results are inside the linear viscoelastic domain, partial complex modulus tests were performed before and after direct traction and compression tests. Exceeding the viscoelastic limit is characterized by a loss of \(|\)E*\(|\). Table 2 presents the percentage of loss of \(|\)E*\(|\) at test temperatures and frequencies. These percentages were calculated using Equation (18). The non-significant losses observed confirm that viscoelastic limit was not exceeded.  
\[\%\ of\ loss=\frac{|E^{\star}_{before\ direct\ creep\ tests}|-|E^{\star}_{after\ direct\ creep\ tests}|}{|E^{\star}_{before\ direct\ creep\ test}|} \tag{18}\]  
The application of the time temperature superposition principle (TTSP) allows having a unique curve. The WLF equation is widely used to determine the shift factor (a\({}_{\text{t}}\)) for bituminous materials. However, this shift factor is used in the frequency domain. In order to apply the TTSP to creep compliance results, two methods are used to determine the shift factor in time domain. First a graphical method is used (experimental: EXP) and second the inverse of a\({}_{\text{t}}\) WLF calculated from complex modulus test were used. The results are presented in Figure 12. It can be noted that the graphical exp shift factor can be approximated by the inverse of WLF equation.  
Figure 11: (**a**) Results of direct tensile and direct compression creep tests and (**b**) error analysis.  
The master curve built with the experimental points is presented at 0 \({}^{\circ}\)C (Figure 13) comparatively with the creep compliance predicted with GKV model and 2S2P1D model coupled to Kopelman approximation. Both of models seem to describe very well the experimental points. These results confirm that it is possible to predict creep compliance (static load test) results directly from complex modulus (cyclic load test) results.