# Elastic Slow Dynamics in Polycrystalline Metal Alloys
## 2 Relaxation Process  
We show here a typical result of a slow dynamics experiment, to summarise the observed phenomenology and to highlight the questions which we aim to discuss in this paper. In these experiments (see Section 3), the propagation velocity \(c_{pre}\) is measured in the relaxed sample (preconditioning) using a low amplitude probing wave. After conditioning, induced using a high strain perturbation, the same low amplitude probing is used to monitor in time the evolution of wave velocity \(c(t)\) (during relaxation).  
A plot as a function of time of the relative velocity variation  
\[\delta c/c=\frac{c(t)-c_{pre}}{c_{pre}} \tag{1}\]  
describes the evolution of the system properties during the relaxation phase and normalizes the velocity variation with respect to differing material properties as in [33]. Since we are considering longitudinal resonances, the relative velocity variation corresponds exactly to the resonance frequency variation \(\delta f/f\) used to monitor slow dynamics in previous works [33]. Note also that analogous considerations could be made for the evolution of damping [14].  
Typical observations are reported in Figure 1 for two metallic samples (Al and Mg based alloys, see Section 3.1) and for a consolidated granular material (mortar). The main features of the relaxation process could be summarised as follows:  
* immediately after conditioning, velocity drops of a quantity \(\Delta c_{cond}\) and an initial not logarithmic in time relaxation occurs for a short time;
* afterwards, in the intermediate time range, the modulus evolves logarithmically with time: \(\delta c/c\propto log(t)\);
* finally, more slowly, velocity relaxes back to its relaxed value: \(\delta c/c\to 0\).  
From the qualitative point of view, strong similarities in the behavior are present, despite the intrinsic different nature of the grain contacts between consolidated granular media (in which the volume between grains is occupied by a consolidating material) and metal alloys (in which there is no loose contact between grains, since the grain boundary in polycrystalline materials is formed by the mismatch between neighbouring crystal lattices). The main features of the relaxation process, which are those normally considered in the literature, i.e., its logarithmic behavior at intermediate times and its long duration in time, are manifested by all samples considered. In all cases full relaxation occurs and, from a qualitative point of view, the duration of the relaxation process is comparable.  
The definition of the time at which relaxation starts (\(t=0\)) is crucial in view of understanding the evolution and identifying the transition to the logarithmic phase. In our approach, we define \(t=0\) as the time corresponding to the switching off of the conditioning amplitude. Electronics delays are negligible, but reverberation of the conditioning phase takes place, during which conditioning and relaxation are mixed-up. This means that the effective initial time of relaxation could be any time between \(0\leq t_{0}\leq t_{rev}\), where \(t_{rev}\) is the reverberation time, which is of the order of tens of milliseconds. However, a time shift of the order of magnitude of \(10^{-2}\) seconds does not affect the analysis presented here and thus the choice adopted for \(t=0\) allows to consider our results reliable and robust.  
Moreover, the definition of the equilibrium velocity (i.e., the value of velocity at \(t\rightarrow\infty\)) is important. As given in Equation (1), the equilibrium velocity at the end of the relaxation process was estimated by the value of wave velocity measured before conditioning \(c_{pre}\), which corresponds to the expected full recovery. This hypotheses was verified to avoid inconsistencies in our fit due to the presence of an unknown offset with respect to \(c_{pre}\). For all datasets, a preliminary fit was performed considering an additional free offset parameter in the fitting function, which was indeed found to be close to zero (with relative values of the order of \(10^{-5}\)).