Figure 1. Different time scales of slow dynamics observed in buildings.
a, The drop and recovery of the resonance frequency (bottom panel) during a single earthquake (top panel). Red line, the co-seismic value of the resonance frequency extracted from the time–frequency distribution diagram. b, Hysteretic recovery during a sequence of aftershocks of the Tohoku earthquake (bottom panel). Each symbol indicates the co-seismic frequency computed during an earthquake, where the stars correspond to significant aftershocks. The solid blue line represents the backbone recovery. The maximum acceleration at the top of the building (PTA) for the aftershocks sequence is also shown (top panel). c, Long-term frequency recovery (bottom panel) during randomly spaced earthquakes (top panel). The event of Aug/2005 was a large-amplitude earthquake (~330 cm/s2), whereas the events shown from Oct/2005 to July/2007 were of the same order of lower amplitude (i.e., the PTAs did not exceed 10% of the large-event PTA).
Backbone recovery curve and hysteresis during aftershocks
Figure 2 shows the co-seismic fundamental frequencies of the ANX and THU buildings, between August 2005 and September 2012 (Fig. 2a). We observe slow dynamics over time following three significant earthquakes (i.e., 2005 Mw 7.2 Miyagi; 2008 Mw 6.9 Iwate; 2011 Mw 9.0 Tohoku; Fig. 2, R1, R2, R3, respectively). We analyzed the time scales of the recoveries of the backbones for the weakest events, which corresponded to the weakest loading, to remove conditioning effects. Assuming a time–logarithmic function28 (Fig. 2c), we observe that the recovery slopes increased with the loading amplitude and the damage state, as also seen previously in several laboratory-tested materials2,4,28. The THU building was exposed to significantly higher levels of maximum acceleration at the top of the building (PTA) and showed recovery slopes that were an order of magnitude larger than for the ANX building before 2011 (i.e., R1, R2). On the other hand, the recovery slope after the Tohoku earthquake (i.e., R3) was around 5-fold steeper for the THU building, which was severely damaged during this event20. Although the log–time adjustment does not have any physical basis, we assume that the rate of recovery is linked in some way to the rate of coalescence within the particles in cracked zones, so that an equilibrium state can be reached. Here, densely cracked media would show steep recovery slopes because there are more voids to be filled after strong excitation.
To explore the conditioning effects on the recovery slope, the log–linear model was applied to the internal recovery cycles created during the aftershock sequence of the Tohoku earthquake (i.e., Fig. 2b, R3a, R3b, and eventually R3c). The recovery slope decreased progressively as the conditioning effects were lost: from 0.069 to 0.057 for ANX, and from 0.196 to 0.128 for THU (Fig. 2c). This suggests a gradual closing mechanism of cracks that were activated by the contribution of local stress-chain adjustments to the total recovery. Furthermore, for the ANX building (which was slightly damaged during the 2011 event20) the recovery slopes due to the conditioning cycles were steeper than the backbone slope, whereas the opposite was seen for the THU building. This reflects the sensitivity of the structural material to the opening/ closing processes of temporary cracks while the structure is still recovering from the main shock. In a densely damaged medium, much more energy would be necessary to perturbate the bond system and generate new stress states that can change the global response, which will be limited, however, by the ultimate collapse of the building.