## 4 Dynamic Tests  
### Measurement of the Frequency Response Functions (FRFs)  
First, the specimens used for complex modulus tension-compression test were sawed to separate the glued upper and lower metallic caps before performing the dynamic measurements. An impact hammer equipped with a load cell (PCB model 086E80) was used as an external source of excitation. The order of magnitude of the maximum strain induced in the specimen by the impact is of about 0.1 \(\mathrm{\SIUnitSymbolMicro m}\)/m [18; 21]. The impact hammer was automated with a solenoid piston programmed with a microcontroller (Arduino Uno R3) to improve the repeatability of the test and to allow measurements directly inside a thermal chamber. This automated system was inspired by systems previously developed by [PERSON] et al. in 2012 and [PERSON] et al. in 2015 [35; 36]. The response of the materials was recorded with an accelerometer (PCB model 353B15). The impact hammer and the accelerometer were connected to a signal conditioner (PCB model 482C15) and the signal conditioner was connected to a data acquisition device (NI USB-6356) connected to a computer. To achieve free boundary conditions, soft foam was placed under the specimens during the tests. In this study, only the longitudinal compression mode of vibrations was considered. For this mode of vibrations, the impact is applied in the centre of one short side of the cylinder while the acceleration is measured in the centre of the opposite short side. The experimental set up for the dynamic tests corresponding to the longitudinal mode of vibrations is presented in Figure 2.  
The measurements were recorded with a sampling frequency of 1 MHz by using a MATLAB application which was specifically developed for this test. Measurements were performed at five temperatures (\(-\)20, 0, 15, 35 and 50 \({}^{\circ}\)C) and five impacts were applied at each temperature. The applied force and the acceleration were recorded for each impact. The experimental data in time domain were then converted in frequency domain with a 1 Hz resolution using the Fast Fourier Transform (FFT). Figure 3 shows an example of the signals in time and frequency domains for specimen GB5-3 at 14.7 \({}^{\circ}\)C (measured temperature with a probe at the surface of the specimen). As shown in Figure 3, the frequency spectrum of the impact contains energy up to 20 kHz, which is the maximum frequency considered for the calculations of frequency response functions (FRFs) in this study. FRFs were calculated from the frequency domain signals as follow:  
\[\mathrm{H(f)}=\left(\overline{\mathrm{Y(f)}\cdot\mathrm{X*(f)}}\right)/\left( \overline{\mathrm{X(f)}\cdot\mathrm{X*(f)}}\right) \tag{8}\]  
where H is the FRF, Y is the FFT of the measured acceleration, X is the FFT of the applied force, X* is the complex conjugate of the applied force and the bar above corresponds to the arithmetic  
Figure 3: Dynamic test experimental data for specimen GB5-3 at 14.7 \({}^{\circ}\)C (5 hits): (**a**) force in time domain; (**b**) acceleration in time domain; (**c**) force in frequency domain; and (**d**) acceleration in frequency domain. Figure 2: Test set up for the dynamic impact tests (example for specimen WF-6, ENTPE laboratory, University of Lyon). average from the five impacts. The five FRFs corresponding to each of the five impacts and the averaged FRF (Equation (8)) for specimen GB5-3 at 14.7 \({}^{\circ}\)C are displayed on Figure 4. Figure 4 shows that the six FRFs overlaps, which confirms the very good repeatability of the test. Quality of the measurements was also checked with the coherence function. The coherence is a value between 0 and 1 that indicates how much of the vibratory response recorded with the accelerometer is due to the impact. For a value of 1, the response is fully explained by the impact while decreasing values indicate that something has disrupted the test.