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.
τmax/τmin) 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.