Introduction
Damage is understood as any change in the material of a system that negatively affects its current or future performance, meaning a loss of its optimal and original design1. All damage begins at the scale of the material, usually as a small defect, an anomaly or heterogeneity of variable degree. Under proper loading conditions, these micro heterogeneities, altogether called ‘the bond system’2, might coalesce until macroscopic alterations are manifested on the scale of the system2-4. Moreover, damage can accumulate gradually over long time periods (i.e. aging effects, cycling loadings, etc.) or can result from unexpected extreme events, such as earthquakes5. Structural health monitoring methods are deployed to detect damage, either based on the continuous assessment of information on system performance (in the case of long-term monitoring), or on the rapid provision of information on system capacity in the case of extreme events. One strategy to monitor structural health of civil engineering structures, consists in tracking variations in dynamic features related to structural capacity, such as stiffness or dissipation coefficients. On the other hand, elastic properties are usually tracked during dynamic excitation of granular material samples2-4, 6-10 and also over seismological studies on fault-systems and crustal areas affected by earthquakes11-12. The common invariant-scale observation is a peculiar non-linear elastic response characterized by a rapid modulus perturbation that is followed by a slow relaxation accompanied by hysteresis and discrete memory2-4. It is believed that this relaxation (i.e. recovery, or slow dynamics) might reflect the state of the bond system. Recent studies have also analyzed non-linear elastic responses of buildings during earthquakes13-14, detecting short- to long-term transitory variations of their resonance frequencies, in relation to the elastic features of the buildings. Is it therefore possible to discern permanent damage from the recovery of elastic properties in buildings? Are theoretical relaxation models able to describe different levels of structural damage, could they be used as a method to monitor seismic structural degradation? How long does it take the recovery of resonance frequencies in buildings affected by earthquakes? How is the slow dynamics recovery of aftershocks following the Tohoku 2011 earthquake?
The article is organized as follows: Section 2 briefly presents the data analyzed and methodology. Section 3 shows the results of monitoring resonance frequencies in buildings during earthquakes, with focus on the long-term frequency recovery after large earthquakes. Relaxation models are then adapted to discern permanent global damage. Final remarks are presented in Section 4.
Data and methodology
It is known that structural damage caused by earthquakes produces permanent frequency changes related to a loss of stiffness13,15-17; this is usually linked to the disconnection of structural and nonstructural elements, joint deformation, variations in the friction/border conditions between elements, and the opening of cracks. The recovery process observed after earthquakes would reveal the (partial or total) restoration of these effects. Here, 7 years of triggered earthquake data were analyzed in two Japanese buildings, namely the ANX and the THU. Both correspond to steel-framed reinforced concrete structures of 8 and 9 stories, respectively. Located in Tohoku, the THU building has faced considerably high loading amplitudes during its lifespan, reaching severe damage during the 2011 Mw. 9.0 event18-19. The ANX building, located in Tsukuba, was slightly damaged during that earthquake20. Besides the 2011 Tohoku event, other important earthquakes were also included in this study (i.e. the Mw 7.2 Miyagi in 2005, the Mw 6.9 Iwate event in 2008). A thorough description of the whole seismic database is available in the Ref. 21. Using the accelerograms from the sensors at the upmost floor, variations in the fundamental frequencies were monitored over time by computing the Wigner-Ville time-frequency distribution for each earthquake, as described in Ref. 13 and 14. The co-seismic frequencies shown along the article correspond to the minimum value of fundamental frequency observed along each Wigner-Ville curve. Peak Top Acceleration (PTA) is the maximum absolute value of acceleration observed at the top floor recordings, used as a proxy of the earthquake loading. To monitor the backbone recovery curves during a sequence of events, only ‘small’ events were considered. An event is considered ‘small’ if its PTA is up to 1% of the maximum PTA observed through the history of the building. Prior to Tohoku, for the THU building, 3% of PTA was used instead, which represent the lowest loading amplitudes for the period.
The log-linear recovery slopes were computed with a first-degree polynomial given by Equation (1):
y = b × log(x) + c (1)
where b is the slope, x is the time, and y is ∆f/ff(Normalized frequency variation, with ∆f/ff = (f-ff)/ff , where ff = maximum final frequency).
Three theoretical models were used to study the long-term recovery processes. These models22-24 are based on physical concepts and were developed based on laboratory experiments that were carried out on the recovery of broken contacts in granular materials. The ratios between final and initial relaxation times (i.e. τmaxmin) were computed from the relaxation function22; whereas the spectral shapes were obtained from the relaxation spectrum model23, and the spectral bandwidth was defined using the 1/√2 of the maximum spectrum amplitude. The procedure to adapt these models22,23 to our data and to compute the relaxation parameters is well described in the Ref. 14, and therefore it will not be explained here. From the long-term relaxation model proposed by the Ref. 24, the parameters a and G were obtained by non-linear regression of Equation (2):
\(\frac{f}{f_{f}}=a\ \log_{10}\left[10^{m}+e^{-G}(1-10^{m})\right]\)(2),
where \(a=\frac{2.3\Lambda}{2E_{o}}\),\(G=\left|\frac{E_{s}}{\Lambda}\right|\), and m described the normalized relaxation times, \(m=\log\left(\frac{t}{\tau}\right)\). The constant \(\Lambda\) is related to loading and E0and Es represent the pre-seismic and co-seismic elastic moduli, respectively. See Ref. 24 for a detailed description.
Observations in Japanese buildings
The multi-scale feature of frequency recovery in buildings
The co-seismic opening of pre-existing cracks might cause the transient material softening at different time scales (Fig. 1). This is shown by the rapid co-seismic decrease of the building’s resonance frequency that is immediately followed by its slow recovery (bottom plot of Fig. 1a). If the earthquake did not cause damage, the pre-earthquake properties of the building will be fully recovered. This reflects the coalescence over time of the granular particles within the perturbed material, towards an equilibrated arrangement2-3 that results in the closing of cracks. During this process, several thermodynamic and mechanical factors control the number of particles in contact within the cracks over time, and consequently, the duration of the recovery3,22.
Over months after a large earthquake (Fig. 1c), we can observe slow recovery over a long-time scale (i.e., of the order of several months, to a few years), in the manner of long-term relaxation of the crustal properties of the Earth after large earthquakes11,12. Strong shakings can open cracks, which might gradually close due to frictional contact between the particles in the damaged zones. Equivalent shaking caused by later smaller earthquakes might contribute to the growth of these contacts, to increase the pressure and friction between the grains, and consequently to favour the recovery process. The recovery of the elastic properties, however, can also be affected by conditioning effects4,7. This is observed in Figure 1b, where the slow dynamics were accompanied by hysteresis and discrete memory during the aftershock sequence of the 2011 Mw 9 Tohoku earthquake. The origin of these effects is in the bond system2-4, and particularly in the spatial arrangements of stress chains25-27, which represent groups of multi-size contacts that relay the strongest stresses. Structural cracking generates stress-chain rearrangements that represent the mechanism for energy dissipation during each event. The energy dissipation depends on the excitation amplitude: small events generally correspond to variations of local stress chains, whereas larger events can cause changes at a global scale, which results in a new complex anisotropic network of cracks that dominates the backbone recovery (i.e., the outer loop) shown in Figure 1b. Internal recovery cycles (i.e., hysteresis) are due to local stress changes that are generated by the strongest aftershocks, without any changes to the general response of the system, and thus with maintenance of the backbone (i.e., the discrete memory). In Figure 1b, the backbone, therefore, describes the recovery of the structural state, which is controlled by the maximum co-seismic strain state of the main shock.