# Elastic Slow Dynamics in Polycrystalline Metal Alloys
## 1 Introduction  
Slow dynamics is perhaps the most characteristic feature of elastic hysteresis manifested in media with a grain structure [1; 2; 3; 4]. These materials exhibit memory effects and hysteresis in their response even when subjected to a low amplitude dynamic excitation (strain of the order of \(10^{-6}\)). Typically, this occurs in rocks [5; 6; 7; 8], concrete [9; 10], mortar [11] or unconsolidated granular media [12]. The phenomenon of slow dynamics consists in the existence of equilibrium states of the material, characterised by given viscoelastic properties (modulus [13] and damping [14]), which are dependent on the strain applied to the material. The relaxation to such equilibrium state is a long-time relaxation process, similar to other phenomena unrelated to the field of elasticity [15; 16; 17; 18]. More specifically, slow dynamics effects consist in a conditioning and a relaxation phase. Given a fully relaxed sample (with unperturbed values of modulus and damping), an external perturbation is applied producing a strain. This phase is called conditioning and the conditioning strain could be generated with a propagating elastic wave [19], by heating/cooling a sample [20; 21] or with an impact [22]. As a consequence, microstructural reorganisation occurs and wave velocity changes in time, following the change in time in modulus which evolves towards its new equilibrium value [23; 24]. The physical mechanisms taking place are still unclear: fluids redistribution [7; 25], sliding, adhesion and/or friction [26; 27; 28], dislocations rearrangement [29] or clapping surfaces in cracks [30], etc. As soon as the conditioning excitation is removed, the system relaxes back, which means that the modulus recovers slowly to its original unperturbed value [31; 32]. The process can be monitored by tracking the evolution of wave velocity in time using a very low amplitude excitation. Relaxation was studied extensively in consolidated granular media or damaged composites. Much less attention was devoted to the behavior of polycrystalline metals and metal alloys. Some evidence of hysteresis was shown in [29] and slow dynamics wasobserved in sintered steel [33], metal alloys [34; 35] and metal monocrystals [36]. More recently, slow dynamic effects were observed in metallic spheres [37]. The reason for the limited attention to elastic hysteresis in metals is mainly due to the fact that effects to be measured are much smaller than in consolidated granular media. Velocity variations of the order of \(10^{-7}\) must be detected to fully capture the details of the relaxation curve. In order to study metallic alloys, two main problems should be considered. First, an experimental set-up with high sensitivity and reliability is needed. Standard approaches extracting velocity evolution monitoring the dependence of the resonance curve on time [38; 39; 40] have to be revised, as shown in Section 3, to make them suitable for studying metals. Here we will discuss an efficient approach based on a semi-analytical inversion (MoDaNE approach) [41; 42], which is an alternative to the more difficult to implement Dynamic Acoustoelastic Testing approach used efficiently by other authors for studying early times relaxation in rocks [43]. The second issue is that of developing efficient post-processing tools which allows modelling the behavior and extracting quantitative information. Indeed, as we will discuss in the next section, the qualitative behavior of polycrystalline alloys looks very similar to that of consolidated granular media, even though the boundaries between grains have very different properties than in concrete or rocks. The long time nature of the relaxation process, suggests to study it as a multirelaxation process [44; 45]. Previous attempts to apply this approach for the interpretation of experimental data were based on using a discrete basis of relaxation times [43] or by introducing a lower and upper cut-off for relaxation times [37].