# From Complex Modulus E* to Creep Compliance D(t): Experimental and Modeling Study
## 3 Results and Discussion  
### Complex Modulus Results and 2S2P1D Calibration  
Complex modulus results are presented in Figures 6 and 7. The calibration of the 2S2S1D model was performed in Cole-Cole curve and black space in order to fix the shape parameters (k, h, \(\delta\) and \(\beta\)). Asymptotic moduli E\({}_{00}\) and E\({}_{0}\) were fixed by a linear extrapolation respectively at very low frequencies and very high frequencies. Characteristic time \(\tau\) depends on temperature as follows:  
\[\mathrm{a_{T}}=\frac{\tau}{\tau_{0}} \tag{8}\]  
where \(\tau_{0}\) is the characteristic time at the reference temperature (T\({}_{\mathrm{ref}}\)) and \(\mathrm{a_{T}}\) is the shift factor. \(\mathrm{a_{T}}\) values may be approximated by the Williams-Landel-Ferry (WLF) equation (Equation (9)) when Time-Temperature superposition is valid [22; 23; 24].  
\[\mathrm{log}(\mathrm{a_{T}})=\frac{-C1\ (T-Tref)}{C2+(T-Tref)} \tag{9}\]  
where C1 (23.90) and C2 (148.90) are empirical constants depending on the material. The calibration of the GKV model was performed according to the minimization proposed by [19; 20], 19 branches of Kelvin-Voigt model were used.  
Master curve is presented in Figure 7 at T\({}_{\mathrm{ref}}\) = 0 \({}^{\circ}\)C. As it is noted, the 2S2P1D model and GKV model describe very well the experimental points. The error between the experimental points and 2S2P1D model are presented in Figure 7. The majority of the points are located within an error interval of \(\pm\) 5% for the norm of complex modulus (\(\mathrm{|E^{*}|}\)) and \(\pm\) 2\({}^{\circ}\) for phase angle (\(\varphi\)). The maximum error noted is 10.5% for \(\mathrm{|E^{*}|}\) and 3.3\({}^{\circ}\) for \(\varphi\). The maximum error between the GKV model and the 2S2P1D model is 14% for \(\mathrm{|E^{*}|}\) and 3\({}^{\circ}\) for \(\varphi\).  
Figure 4: Sample preparation for complex modulus test, direct tensile test and direct compression test.  
Figure 5: Sample preparation for indirect tensile test.  
Figure 6: Complex modulus results in (**a**) Cole–Cole representation and (**b**) black space.  
The calculation of creep compliance is done using two different methods. First, using the GKV model, the creep compliance of this model is given by Equation (6). The second method consists of coupling the 2S2P1D results with the approximation proposed by [PERSON]. The 2S2P1D model doesn't have a creep function in frequency domain while using Kopelman Equation (15) can allow to switch from the frequency domain to the time domain. The explanations of the calculations are given below:  
Figure 7: (**a**) Master curve of complex modulus. (**b**) the error between 2S2P1D model and the experimental points.  
\[E^{*}=E_{1}+iE_{2} \tag{10}\]  
\[D^{*}=D_{1}-iD_{2} \tag{11}\]  
\[E^{*}(\omega)\times D^{*}(\omega)=1 \tag{12}\]  
\[D_{1}=\frac{E_{1}}{E_{1}^{2}+E_{2}^{2}} \tag{13}\]  
\[D_{2}=\frac{E_{2}}{E_{1}^{2}+E_{2}^{2}} \tag{14}\]  
\[D(t)=|D(\omega)|_{\omega=1/t} \tag{15}\]  
where:  
E\({}_{1}\): the real part of the complex modulus (storage modulus) (MPa).  
E\({}_{2}\): the imaginary part of the complex modulus (loss modulus) (MPa).  
D\({}_{1}\): the real part of the complex compliance (1/MPa).  
D\({}_{2}\): the imaginary part of complex compliance (1/MPa).  
i: the imaginary unit.  
Creep compliance from the GKV model and from the 2S2P1D model coupled with Kopelman approximation are presented in Figure 8. The two curves are very close. The error is calculated from \(10^{-5}\) s to \(10^{+5}\) s using Equation (16).  
\[\%\ Error=\frac{D(t)_{2S2P1D+Kopelman}-D(t)_{GKV}}{D(t)_{2S2P1D+Kopelman}} \tag{16}\]  
Figure 8: Creep compliance predicted with the GKV model and with the 2S2P1D coupled to Kopelman approximation.  
The results are presented in Figure 9. The error is between \(-2\%\) to \(-15\%\).