## 3 Experimental Results
### Velocity and Damping Dependence on Strain  
Velocity and damping could be derived (using MoDaNE fitting, as discussed in Section 2.3) and plot vs. output amplitude (strain). In the following, data referring to velocity are discussed, but the same applies to damping as well. It is possible to introduce the conditioned and baseline relative velocity variations:  
\[\delta v_{\text{C,B}}=\frac{v_{\text{C,B}}-v_{\text{lin}}}{v_{\text{lin}}}, \tag{4}\]  
where \(v_{\text{lin}}\) is the linear velocity at the given temperature, measured by exciting the sample at a very low excitation amplitude (smaller than or equal to the baseline drive amplitude). Note that Equation (4) applies to both the conditioned and baseline states, thus justifying the notation used for the pedices. The velocity measured at conditioning strain \(\delta v_{\text{C}}\) contains contributions from both fast and slow dynamics. Separating the two terms is desirable, since they might (or might not) originate from different physical mechanisms and, depending on the driving parameter (which, here, is temperature) the relative weight of the two contributions might change. Furthermore, slow dynamics-related effects strongly depend on the experimental set-up properties and, thus, it is advisable to remove them to obtain observations that are only \"sample-dependent\". During conditioning at a given drive amplitude, a contribution to the velocity variation, larger for larger amplitudes, is slowly built up over time. As soon as the excitation is switched to the baseline amplitude, this contribution slowly relaxes back to zero. Therefore, slow dynamics originates a contribution which increases with the duration of the drive [1] and is due to the evolution of the material properties towards a strain-dependent equilibrium state, which is only partially achieved in the \(\Delta t\) defining the duration of the chirp. The slow contribution does not disappear instantly; thus, it contributes to the determination of the velocity measured at baseline (low amplitude) drive. Assuming the relaxation of the slow dynamic contribution before probing the baseline to be small, the fast contribution to velocity variation can be defined as  
\[\delta v_{\text{fast}}=\delta v_{\text{C}}-\delta v_{\text{B}}=\frac{v_{ \text{C}}-v_{\text{B}}}{v_{\text{lin}}}. \tag{5}\]  
In Figure 3, the strain dependence of conditioned, baseline, and fast velocity variations are reported compared to the strain (output amplitude). The results show some common features in the behavior of the velocity variation vs. strain for Berea and concrete:  
* \(\delta v_{\text{C}}\): hysteresis is present, given by the difference between the loading and unloading branches;
* \(\delta v_{\text{B}}\): besides the hysteresis (which is even more remarkable than in the conditioned data), it can be observed that the maximum baseline velocity variation does not occur when the conditioning strain is maximum; see the further decrease in velocity observed during the first instances of unloading (blue symbols);
* \(\delta v_{\text{fast}}\): the hysteresis, and hence, slow dynamic effects, is completely removed. However, apparent changes in the functional form of the dependence can be observed. At first glance, a quadratic dependence seems to be shared by all curves, but with positive/negative quadratic terms for concrete/Berea. All data (except unloading data for concrete) also seem to present a strain interval in which the behavior is linear. This wealth of behaviors is indeed reflective of the complexity of NRUS data, which can be found in the literature. At least for Berea, depending on the strain testing range, different authors have observed quadratic, linear or S-shaped behaviors [1; 42; 43; 44; 45]. It is our assumption here that the latter might be a better representation of the velocity dependence on strain.