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.