Figure 2 Long-term recovery of the fundamental frequency of the buildings (ANX, THU).
a, Co-seismic frequency computed over the years, showing slow dynamics after the important earthquakes in 2005, 2008 and 2011.b, Zoom-in on the recovery after the 2011 event, showing the conditioning cycles (i.e., R3a, R3b, R3c). a, b, Each symbol corresponds to a single earthquake, and the colour scale is related to the PTA. The larger symbols were used to monitor the backbone curve. c, Log–linear recovery of the normalized frequency variation (∆f/ff = (f-ff)/ff , where ff = maximum final frequency), which indicates the slopes computed from the log–linear method applied to the periods shown in a andb .
Relaxation models applied to long-term structural recovery
Firstly, from the relaxation model24 shown in Figure 3a, the a and G parameters were computed as proxies for the elasticity. The parameter a , which is inversely proportional to the pre-seismic elastic modulus, increased sharply during the post-Tohoku recovery. Parameter G, which is directly proportional to the co-seismic elasticity, decreased. This confirms the increase in the softening in both of these buildings. Secondly, the ratio τmaxmin computed from the relaxation function in Figure 3b22 increased from ~6 to ~23 in the ANX building, and from ~9 to ~18 for THU. This ratio denotes the different time-scale mechanisms that act in the time–logarithmic segment of the recovery and characterizes the diversity of the crack sizes14. From this we can infer that after the 2011 event the variety of the cracks in the ANX and THU structures was quadrupled and doubled, respectively. Additionally, the gradual reduction in τmaxmin during the re-loading cycles confirms the progressive crack-closing process during the aftershocks inferred from the recovery slopes. In addition, after the Tohoku earthquake we observe clear changes in the maximum frequency variation (i.e., ∆f/f) for both models, which increased from ~3% to ~12% for ANX and from ~13% to ~33% for THU (Fig. 3a, b), which is consistent with the modulus softening and then the global change in the structural states.
A complete signature of the recovery process is given by the relaxation spectrum23 shown in Figure 3c. We detect mechanisms over five orders of magnitude in time, from t ~0.1 to ~1200 seconds; i.e., extreme values that are not revealed by the previous models. The spectrum bandwidth represents the range of the dominant relaxation times, and as the ratio τmaxmin, this serves as a hint of the diversity of the crack sizes that are closed over the time of the recovery. These data suggest that a large variety of crack sizes was activated following the 2008 event (i.e., R2). The post-Tohoku spectrum does not indicate new types of cracks; nevertheless, the maximum spectrum amplitude is ~3.5-fold that observed in the periods prior to Tohoku. This implies that the crack density increased around 3.5-fold after 2011 for both of these structures. It can also be noted that the spectra of the THU building are approximately 3-fold those of the ANX building, showing the different levels of damage between the buildings even before 2011. At the same time, conditioning effects might have been significant in the recovery process for the ANX building, which activated mechanisms with relaxation times in the order of 101-102 seconds. In contrast, during the recovery of the THU structure, the conditioning cycles just contributed to the activation of inner small mechanisms, as shown by the narrow left-shifted spectra R3a,b,c (Fig. 3c, right). Thus, theoretical models applied to earthquake data from real buildings fit the recovery of the fundamental frequency after earthquakes. These data indicate that non-linear elastic processes within the structural bond system might explain the transitory and permanent variations of structural dynamic responses to seismic events. In particular, the relaxation parameters reveal the internal material changes that are related to cracking and stiffness degradation; i.e., in relation to the structural health and safety of a building.