applied sciences

Article Fluctuations of 1/f Noise in Damaging Structures Analyzed by Acoustic Emission

Alberto Carpinteri, Giuseppe Lacidogna * and Federico Accornero

Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, 24, Corso Duca degli Abruzzi, 10129 Torino, Italy; [email protected] (A.C.); [email protected] (F.A.) * Correspondence: [email protected]; Tel.: +39-011-090-4971



Received: 17 August 2018; Accepted: 6 September 2018; Published: 18 September 2018


Abstract: It is well known in literature that frequency fluctuations of different physical quantities clearly show 1/f noise power spectra. In the present work, the authors observe that in some brittle materials, such as concrete, masonry, and mortar, Acoustic Emission (AE) signals, generating from brittle fracture phenomena, exhibit a frequency fluctuation approaching to 1/f. Acoustic Emission data obtained from laboratory tests on concrete samples, and from in-situ monitoring of some important Italian historical buildings are reported in terms of spectral density vs. frequency. It is shown that in structural elements subjected to different load conditions, the frequency fluctuations are 1/f like. The study and interpretation of these phenomena through the use of the AE technique can be therefore very useful for identifying the transition from the critical conditions of a structure to those that involve an incipient collapse.

Keywords: 1/f noise; Acoustic Emission; damage monitoring; self-organized criticality; architectural heritage

1. Introduction The presence of power laws in critical phenomena [1,2] seems to indicate that something deep appears in the spectral density of the so-called flicker noise, that is actually rather variable. It runs like 1/f γ, where 0.5 ≤ γ ≤ 1.5, over different frequency magnitude orders. In particular, 1/f noise is associated to a signal whose spectrum is linear in a logarithmic scale. A color name, “pink noise”, is given by analogy with pink light, which has such a linear power spectrum over the visible range [1]. More generally, different colors are associated to noise with spectral profiles other than pink noise (γ = 1), in reference to the light with analogous spectra [3], e.g., the spectral density of white noise is flat (γ = 0), while red (or Brownian) noise has γ = 2. Power laws or 1/f spectra were found very unexpectedly in different phenomena, and several authors [1,4,5] are convinced that there had to be an ubiquitous reason for this kind of power-law noises, linked to the universality of critical phenomena exponents. Curiously enough, 1/f noise is present in nature in very unexpected places, i.e., the flow of the river Nile, the luminosity of stars, the loudness of a piece of classical music versus time, the flow of sand in an hourglass, and many others [1]. Clarke and Voss [6] found that both voice and music broadcasts have 1/f spectra, and they proposed an algorithm to compose “fractal music”. A crucial feature of 1/f noise is its scale invariance for any frequency or time range, and for this reason it has been considered as a wide manifestation of the fractal character of several natural phenomena. Moreover, also nonlinear processes can be often considered as sources of 1/f noise [1,4,5].

Appl. Sci. 2018, 8, 1685; doi:10.3390/app8091685 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 1685 2 of 14

After Mandelbrot’s work [2], 1/f noise has been linked to fractal phenomena and power laws. The concept of scaling, which leads to power laws, is present also in the physics of earthquakes [7–9]. This field has its cornerstone in the Gutenberg-Richter (GR) law [10], i.e., Log10N(≥ m) = a − bm, or N(≥ m) = 10 a−bm. This equation expresses the relationship between magnitude and total number of earthquakes with the same or a higher magnitude, in any given region and time period, and it is the most widely used statistical relation to describe the scaling properties of seismicity. N is the cumulative number of earthquakes with magnitude ≥ m in a given area and within a specific time range, whilst a and b are positive constants varying from a region to another and from a time interval to another. Richter’s law has been successfully employed also within the Acoustic Emission (AE) technique to study the scaling laws of signal amplitude distribution [7–9]. According to Richter’s law, the b-value systematically changes at different times during the damage process. Therefore, it can be used to estimate the modalities of fracture evolution [11]. To establish a relationship between the size L of the defect associated to an AE event, and its magnitude m, the Richter’s law can be rewritten: N(≥ L) = cL−2b, where 2b = D is the fractal dimension of the damage domain, c is a constant of proportionality, N is the cumulative number of AE events generated by source defects with a characteristic linear dimension ≥ L. The implications of the assumption 2b = D has been pointed out by Aki [12], Main [13–15], and Carpinteri [16,17], assuming that 2.0 ≤ D ≤ 3.0, i.e., the cracks are distributed in a fractal domain comprised between a fracture surface and the volume of the structural element [18–21]. In the present work, the authors have observed that even in quasi-brittle materials, like rocks, concrete, and masonry, there is a source of 1/f noise in AE from fracture phenomena. The results of laboratory tests on specimens of rocks and concrete, and of in-situ monitoring of Italian historical buildings are considered. The obtained AE data are represented in a log-log plot of spectral density versus frequency, showing a 1/f noise behaviour of pressure waves in structural elements subjected to different loading conditions. Finally, a discussion on 1/f statistical distribution of AE events and the so-called b-value variations in the performed tests will be conducted. Briefly, in the following a list of symbols is provided in order to better understand either the text or the figures in the present context: f : Acoustic Emission signal frequency; γ: frequency exponent characterizing the power spectral density; b:(b-value) scaling parameter estimating the number of AE signals, NAE(A), with an amplitude equal or greater than A; D: fractal damaged domain comprised between a volume and a surface.

2. 1/f Noise in Damaging Structures: Laboratory Tests Acoustic Emission (AE) is a Non-Destructive Inspection Technique that utilizes the transient elastic waves produced by fracture phenomena. These waves are picked up by piezoelectric sensors positioned on the external surface of the monitored body [22,23]. In laboratory tests and in-situ campaigns, the authors exploited a leading-edge monitoring equipment [24] for multichannel data processing, and consisting in six synchronized “Units for Storage Acoustic Emission Monitoring” (USAM). Each of these units analyzes in real time, and transmits to a computer, all the characteristic parameters of an ultrasonic event. Specific waveform parameters are set as in the following: Peak Definition Time (PDT) = 50 µs; Hit Definition Time (HDT) = 150 µs; Hit Lockout Time (HLT) = 300 µs. In this manner, each AE event is identified by a progressive number and characterized by a series of data giving the amplitude and time duration of the signals and the number of oscillations. Moreover, the absolute acquisition time and signal frequency are also given, so that, through an analysis of signal frequency and time of arrival, it is possible to identify the group of signals belonging to the same AE event and to localize it. Each unit is equipped with a pre-amplified Appl. Sci. 2018, 8, 1685 3 of 14

Appl.wide-band Sci. 2018, piezoelectric 8, x sensor, which is sensitive in a frequency range between 50 kHz and 8003 kHz.of 14 The signals acquisition threshold can be set in a range between 100 µV and 6.4 mV. In this this section, section, the the results results of compression of compression tests performedtests performed on concrete on concrete specimens specimens having diameter having diameterequal to 100 equal mm, to and100 mm, slenderness and slendernessλ equal to λ 1.0 equa andl to 2.0 1.0 are and presented 2.0 are presented (Figure1). (Figure Generally 1). Generally speaking, speaking,in a frictionless in a frictionless condition, for condition, a specimen for a with specimen slenderness with slendernessλ ≤ 1.0 a failure λ ≤ 1.0mainly a failure governed mainly by governedcrushing isby expected, crushing whileis expected, for a specimen while for with a specimen slenderness with λslenderness≥ 2.0 a failure λ ≥ 2.0 governed a failure by governed splitting byis taken splitting into account. is taken For into intermediate account. For values intermediate of slenderness, values shear of slenderness,failure is characterized shear failure by the is characterizedformation of inclinedby the formation slip bands of in inclined the bulk slip of bands the specimen. in the bulk For of all the cases, specimen. the cumulated For all cases, number the cumulatedof AE is strictly number correlated of AE to is global strictly or correlated local mechanical to global instabilities or local mech in theanical post-peak instabilities regime in of the postloading-peak process. regime of the loading process. The loading process was controlled by the specimen circumferential expansion, measured measured by by means of a linkedlinked chainchain placedplaced aroundaround the the sample sample mid-height mid-height (Figure (Figure1b). 1b). Such Such a controla control permitted permitted to tofully fully detect detect the load-displacement the load-displacement curve, curve, even in even case in of case snap-back of snap phenomena-back phenomena [25,26]. The [25,26]. specimen The specimenwith a slenderness with a slendernessλ equal to λ1.0 equal was equippedto 1.0 was with equipped a thin with layer a of thin Teflon layer in ofcontact Teflon between in contact the betweenplaten of the the platen testing of machine the testing and machine the specimen and the ends, specimen to avoid ends, the frictionto avoid on the the friction bases andon the facilitate bases andthe transversalfacilitate the deformation. transversal deformation.

(a) (b)

Figure 1. 1. ConcreteConcrete specimens specimens subjected subjected to compression to compression test monitored test monitored by means by of Acoustic means ofEmission Acoustic (a); EExperimentalmission (a); Experimental set-up (b). set-up (b).

During the compression tests, Acoustic Emission technique was adopted to monitor each specimen. AE AE signals signals generating generating from from the the fracture fracture phenomena phenomena were were detected detected by by attaching a piezoelectric sensor to the surface of the specimens specimens with a a silicon silicon glue. glue. As As stated, the the sensor is sensitive in inthe the frequency frequency range range (50–800 (50–800 kHz), kHz), and it and is placed it is placed on the external on the surfaceexternal of surface the monitored of the monitoredsamples (Figure samples1b). The (Fig AEure acquisition1b). The AE threshold acquisition level threshold was set up level to 2 mV, was with set up an amplification to 2 mV, with gain an amplificationof the AE signals gain equal of the to AE 60 signals dB. The equal recorded to 60 waveformdB. The recorded sampling waveform frequency samp wasling 1 Msample/s. frequency was 1 Msample/s.The results obtained from the compression test of specimen with λ = 1.0 are represented in FiguresThe2 – results4. The obtained signal frequency from the refers compression to the average test of values, specimen the with average λ = frequency 1.0 are represented (AF), which in Figuresis calculated 2–4. The as the signal ratio frequency between therefers AE to ring-down the average counts values, and the the average signal frequency duration time, (AF), while which the is calculatedenergy of eachas the AE ratio signal between is calculated the AE according ring-down to RILEMcounts [and27]. the signal duration time, while the γ energy1/ off eachdistribution AE signal of is AE calculated signals is according divided intoto RILEM two time [27]. windows: before the achievement of the peak1/fγ distribution load, and after of AE the signals peak load is divided (Figure into2). In two Figure time3 ,windows: the log-log before plot of the AE achievement power spectrum of the peakversus load, frequency and after for signalsthe peak detected load (Figure before 2). the In peak Figure load 3, is the represented. log-log plot Therefore, of AE power a fluctuation spectrum of γ versusf - (γ ≈ frequency0.29) is found, for signals which detected reveals abefore source the noise peak basically load is “white”represented. (“white” Therefore, or random a fluctuation noise: 1/ fof0). f-γ (γ ≈ 0.29) is found, which reveals a source noise basically “white” (“white” or random noise: 1/f0). Appl. Sci. 2018, 8, x 4 of 14 Appl.Appl. Sci.Sci.2018 2018,,8 8,, 1685 x 44 of of 14 14

Figure 2. Concrete specimen with λ = 1: Load vs. time diagram; Cumulated AE (hits) versus time FigureFigure 2.2. ConcreteConcrete specimenspecimen withwith λλ= = 1:1: LoadLoad vs.vs. timetime diagram;diagram; CumulatedCumulated AEAE (hits)(hits) versusversus timetime diagram. Red line refers to the peak load occurrence. diagram.diagram. RedRed lineline refersrefers toto thethe peakpeak loadload occurrence.occurrence.

Figure 3. Log-log plot of AE power spectrum versus frequency for the concrete specimen (λ = 1): 1/f γ Figure 3. Log-log plot of AE power spectrum versus frequency for the concrete specimen (λ = 1): 1/fγ noiseFigure before 3. Log peak-log load,plot of with AEγ power= 0.29. spectrum White noise versus source frequency (Random for phenomena).the concrete specimen (λ = 1): 1/fγ noise before peak load, with γ = 0.29. White noise source (Random phenomena). noise before peak load, with γ = 0.29. White noise source (Random phenomena). Figure4 shows the log-log plot of AE power spectrum versus frequency for signals detected after Figure 4 shows the log-log plot of AE power spectrum versus frequency for signals detected the peakFigure load. 4 shows For the the compression log-log plot test of ofAE the power specimen spectrum with λversus= 1.0 afrequency fluctuation for of signalsf -γ (γ ≈dete0.9)cted is after the peak load. For the compression test of the specimen with λ = 1.0 a fluctuation of f-γ (γ ≈ 0.9) found,after the which peak reveals load. For a source the compression noise basically test “pink”of the specimen (“pink” or with correlated λ = 1.0 noise:a fluctuationγ ~ 1.0 of for f- 1/γ (γf γ ≈). 0.9) is found, which reveals a source noise basically “pink” (“pink” or correlated noise: γ ~ 1.0 for 1/fγ). is found,Furthermore, which reveals the results a source obtained noise basically from the “pink” compression (“pink” or test correlated of specimen noise: withγ ~ 1.0λ for= 2.0 1/fγ are). represented in Figures5–7. Also in this case, 1/ f γ distribution of AE signals is divided into two time windows (Figure5): before the achievement of the peak load, and after peak load. In Figure6 the log-log plot of AE power spectrum versus frequency for signals detected before the peak load is reported, and a fluctuation of f -γ (γ ≈ 0.33) is found, which reveals a source noise basically “white” (“white” or random noise: 1/f 0). Appl. Sci. 2018, 8, x 5 of 14

Appl. Sci. 2018, 8, 1685 5 of 14 Appl. Sci. 2018, 8, x 5 of 14

Figure 4. Log-log plot of AE power spectrum versus frequency for the concrete specimen (λ = 1): 1/fγ noise after peak load, with γ = 0.90. Pink noise source (Correlated phenomena).

Furthermore, the results obtained from the compression test of specimen with λ = 2.0 are represented in Figures 5–7. Also in this case, 1/fγ distribution of AE signals is divided into two time windows (Figure 5): before the achievement of the peak load, and after peak load. In Figure 6 the log-log plot of AE power spectrum versus frequency for signals detected before the peak load is Figure 4. λ f γ reported,Figure and 4. Log-logLog a fluctuation-log plot ofof AEAE of powerpowerf-γ (γ ≈ spectrumspec 0.33)trum is versus found,versus frequency frequency which reveals for for the the concretea concrete source specimen specimennoise basically ( (λ= = 1): 1): 1/ 1/“white”fγ noise after peak load, with γ = 0.90. Pink noise source (Correlated phenomena). (“white”noise orafter random peak load, noise: with 1/ fγ0). = 0.90. Pink noise source (Correlated phenomena).

Furthermore, the results obtained from the compression test of specimen with λ = 2.0 are represented in Figures 5–7. Also in this case, 1/fγ distribution of AE signals is divided into two time windows (Figure 5): before the achievement of the peak load, and after peak load. In Figure 6 the log-log plot of AE power spectrum versus frequency for signals detected before the peak load is reported, and a fluctuation of f-γ (γ ≈ 0.33) is found, which reveals a source noise basically “white” (“white” or random noise: 1/f0).

λ FigureFigure 5.5. ConcreteConcrete specimenspecimen withwith λ == 2:2: LoadLoad vs.vs. timetime diagram;diagram; CumulatedCumulated AE (hits)(hits) versusversus timetime diagram.diagram. RedRed lineline refersrefers toto thethe peakpeak loadload occurrence.occurrence.

Figure 5. Concrete specimen with λ = 2: Load vs. time diagram; Cumulated AE (hits) versus time diagram. Red line refers to the peak load occurrence. Appl. Sci. 2018, 8, x 6 of 14 Appl. Sci. 2018, 8, 1685 6 of 14 Appl. Sci. 2018, 8, x 6 of 14

Figure 6. Log-log plot of AE power spectrum versus frequency for the concrete specimen (λ = 2): 1/f γ FigureFigure 6. Log 6. Log-log- logplot plot of ofAE AE power power spectrum spectrum versusversus frequencyfrequency for for the the concrete concrete specimen specimen (λ = 2):(λ 1/= f2):γ 1/fγ noise before peak load, with γ = 0.33. White noise source (Random phenomena). noisenoise before before peak peak load, load, with with γ =γ =0.33. 0.33. White White noisenoise source (Random (Random phenomena). phenomena).

Analogously,Analogously, in Figurein Figure7 the 7 the log-log log-log plot plot of of AEAE powerpower spectrum spectrum versus versus frequency frequency for si forgnals signals Analogously, in Figure 7 the log-log plot of AE power spectrum versus frequency for signals detecteddetected after after the the peak peak load load is is shown. shown. During During the the compression compression test test of specimenof specimen with with λ = 2.0λ = a 2.0 a detected after the peak load is shown. During the compression test of specimen with λ = 2.0 a fluctuationfluctuation of f - γ of( γf-γ≈ (γ0.84) ≈ 0.84) is found, is found, which which reveals reveals a source a source noise noise basically basically “pink” “pink” (“pink” (“pink” or correlated or -γ fluctuation of f (γ ≈1 0.84) is found, which reveals a source noise basically “pink” (“pink” or noise:correlated 1/f 1). noise: 1/f ). correlated noise: 1/f1).

Figure 7. Log-log plot of AE power spectrum versus frequency for the concrete specimen (λ = 2): 1/fγ noise after peak load, with γ = 0.84. Pink noise source (Correlated phenomena). Figure 7. Log-log plot of AE power spectrum versus frequency for the concrete specimen (λ = 2): 1/f γ γ noiseFigureSummarizing, after 7. Log peak-log load, plot the withof frequency AEγ pow= 0.84.er fluctuation spectrum Pink noise dataversus source obtained frequency (Correlated by thefor phenomena). thecompression concrete specimen tests carried (λ = out 2): 1/onf concretenoise after cylin peakdrical load, specimens with γ = 0.84. show Pink how, noise before source the (Correlated achievement phenomena). of the peak load, AE signals Summarizing,reveal a source noise the frequencybasically “white” fluctuation (1/f0) that data is obtainedcharacteristic by theof prominent compression random tests phenomena. carried out on concreteOnSummarizing, the cylindrical contrary, specimenstheafter frequency peak load, show fluctuation the how, source before noisedata the revealedobtained achievement by by AE the ofsignal thecompression peakis characterized load, tests AE signals carriedas “pink” reveal out on a sourceconcrete(1/ noisef1), cylin that basically isdrical distinctive specimens “white” of correlated (1/ showf 0) thatphenomena. how, is characteristic before See, the for achievementthe of sake prominent of synthesis, of random the the peak graph phenomena. load, of Figure AE signalsOn 8 the contrary,revealthat a sourcerepresents after peaknoise the load, basicallybehaviour the source“white” under noisecompression (1/f0) revealed that isof characteristicthe by specimen AE signal having of is prominent characterized slenderness random 1. as “pink” phenomena. (1/f 1), thatOn the is distinctivecontrary,Additionally, after of correlatedb -peakvalues load, of phenomena.the the specimen source with See,noise λ for= revealed1.0 the are sake reported by of AE synthesis, in signal Figure is9 the bothcharacterized graph for pre of-peak Figure as (b “pink” 8≅ that 1.40), and post-peak loading phases (b ≅ 1.02), confirming the 1/f noise as a very suitable indicator of represents(1/f1), that is the distinctive behaviour of under correlated compression phenomena. of the See, specimen for the sake having of slendernesssynthesis, the 1. graph of Figure 8 both the critical states and those that precede global or local collapses [25,26]. As a matter of fact, as it that representsAdditionally, the bbehaviour-values of under the specimen compression with ofλ =the 1.0 specimen are reported having in Figureslenderness9 both 1. for pre-peak ∼ is well known in the literature [7–21], the∼ statistical interpretation to the variation in the b-value (b = 1.40),duringAdditionally, and the post-peak evolution b-values loading of of damage the phasesspecimen detected (b = with 1.02), by λ AE confirming= 1.0 captures are reported the the transition1/ finnoise Figure asfrom a9 very both the suitable conditionfor pre- indicatorpeak of (b ≅ of1.40), bothdiffused and the post critical criticality-peak states loading to that and ofphases those imminent that(b ≅ precede1.02),failure confirming localization. global or the localMore 1/ fcollapsesprecisely, noise as a [in25 very the,26 ].pre suitable As-peak a matter loadingindicator of fact, of asboth it isphasethe well critical of known the statesspecimen in theand with literature those λ = that 1.0, [7 precedeto–21 the], thewhite global statistical noise or envisaged local interpretation collapses by the [25,26].1/ tof analysis, the variationAs acorresponds matter in of the fact, theb-value as it duringis well the known evolution in the of literature damage detected [7–21], the by AEstatistical captures interpretation the transition to from the the variation condition in the of diffusedb-value criticalityduring the to evolution that of imminent of damage failure detected localization. by AE More captures precisely, the transition in the pre-peak from loadingthe condition phase of thediffused specimen criticality with λto= that 1.0, of to imminent the white noisefailure envisaged localization. by theMore 1/ fprecisely,analysis, in corresponds the pre-peak the loading critical phase of the specimen with λ = 1.0, to the white noise envisaged by the 1/f analysis, corresponds the Appl. Sci.Appl.2018 Sci., 82018, 1685, 8, x 7 of 147 of 14 Appl. Sci. 2018, 8, x 7 of 14 critical condition also confirmed by the b-value near to 1.5. In this case the energy emission takes critical condition also confirmed by the b-value near to 1.5. In this case the energy emission takes conditionplace alsothrough confirmed small defects by the homogeneouslyb-value near to distributed 1.5. In this throughout case the energy the specimen’s emission takesvolume. place On throughthe place through small defects homogeneously distributed throughout the specimen’s volume. On the smallcontrary, defects homogeneouslyin the post-peak loading distributed phase, throughout to the pink noise the specimen’senvisaged by volume. the 1/f analysis, On the corresponds contrary, in the contrary, in the post-peak loading phase, to the pink noise envisaged by the 1/f analysis, corresponds post-peaka b-value loading near phase, to 1.0, to in the this pink situation noise the envisaged energy emission by the 1/ takesf analysis, place on corresponds preferentially a b fracture-value near a b-value near to 1.0, in this situation the energy emission takes place on preferentially fracture to 1.0,surfaces in this [9]. situation the energy emission takes place on preferentially fracture surfaces [9]. surfaces [9].

FigureFigure 8. Concrete 8. Concrete specimen specimen (λ (λ= 1)= 1) subjected subjected toto compressioncompression test: test: Load Load vs. vs. time time diagram; diagram; Cumulated Cumulated Figure 8. Concrete specimen (λ = 1) subjected to compression test: Load vs. time diagram; Cumulated AE versusAE versus time time diagram diagram with with signal signal frequency frequency fluctuationsfluctuations (qualitative (qualitative trend). trend). Red Red line linerefers refers to the to the AE versus time diagram with signal frequency fluctuations (qualitative trend). Red line refers to the peak loadpeak load occurrence. occurrence. peak load occurrence.

(a) (b) (a) (b)

Figure 9. Concrete specimen (λ = 1.0) subjected to compression test: b-value for pre-peak loading phase (a); b-value for post-peak loading phase (b). Appl. Sci. 2018, 8, x 8 of 14

Appl. Sci.Figure2018, 8 , 9. 1685 Concrete specimen (λ = 1.0) subjected to compression test: b-value for pre-peak loading 8 of 14 phase (a); b-value for post-peak loading phase (b).

3. 1/3. f1/Noisef Noise in in Damaging Damaging Structures: Structures: In-SituIn-Situ AE AE Monitoring Monitoring AEAE technique technique has has been been veryvery usefullyusefully employed by by the the authors authors to to investigate investigate on on a wide a wide set setof of structuralstructural problems problems related related to toancient ancient masonry masonry buildings buildings [28 [28,29].,29]. In In particular, particular, AE AE technique technique has has been employedbeen employed to assess to theassess stability the stability of the tallestof the tallest structure structure in the in city the of city Bologna of Bologna (Italy), (Italy), the Asinelli the Asinelli Tower, which,Tower, together which, together with the with nearby the nearby Garisenda Garisenda tower, tower, is the is renowned the renowned symbol symbol of the of the city city (Figure (Figure 10 10))[30 ]. The[30]. Asinelli The Asinelli Tower Tower dates backdates toback the to early the early XII Century, XII Centu andry, itand is it the is tallestthe tallest leaning leaning tower tower in Italy.in Italy. The tower measures 97.30 m in height, the side is 8.00 m at the base, and 6.50 m at the top. Its The tower measures 97.30 m in height, the side is 8.00 m at the base, and 6.50 m at the top. Its leaning leaning is of 2.38 m westward. is of 2.38 m westward.

FigureFigure 10. 10.The The GarisendaGarisenda Tower and the Asinelli Asinelli Tower, Tower, in in Bologna Bologna City City center. center.

TheThe AE AE activity activity was was detected detected in in a a masonry masonry significant significant region region forfor monitoringmonitoring purposespurposes and easy to reach.to reach. Six piezoelectric Six piezoelectric sensors sensors (working (working in the in range the of range 50–800 of 50 kHz)–800 were kHz) attacked were attacked on the north-east on the cornernorth of-east the corner tower, of immediately the tower, imme abovediately the above terrace the atop terrace the atop arcade, the atarcade, an average at an average level of level ~9.00 of m above~9.00 ground. m above In ground. this area, In the this double-wall area, the double masonry-wall has masonry a thickness has a of thickness ~2.45 m. of AE ~2.45 monitoring m. AE beganmonitoring on 23 Septemberbegan on 23 2010 September at 5:40 2010 p.m., at and5:40 endedp.m., and on ended 28 January on 28 2011January at 1:002011 p.m.,at 1:00 covering, p.m., therefore,covering, a therefore, 4-month period. a 4-month The period. AE monitoring The AE monitoring was carried was on carried exploiting on exploiting the above the mentioned above USAMmentioned system. USAM To filter system. out theTo filter environmental out the environmental background background noise a detection noise a threshold detection ofthreshold 100 mV of was settled100 mV (appropriate was settled for (appropriate masonry accordingfor masonry to according the authors’ to the experience authors’ experience [28,30]). From [28,30]). the monitoringFrom the processmonitoring carried process out on carried a significant out on a part significant of this tower,part of itthis was tower, possible it was to possible evaluate to the evaluate incidence the of seismicincidence activity, of seismic vehicle activity, traffic, vehicle and wind traffic, action and wind on fracture action evolutionon fracture and evolution damaging and damaging phenomena withinphenomena the structure. within the structure. In Figure 11, AE data obtained during the structural monitoring are represented in a log-log In Figure 11, AE data obtained during the structural monitoring are represented in a log-log plot plot of power spectrum versus frequency, with a fluctuation of f-γ (γ = 0.94). Therefore, the source of power spectrum versus frequency, with a fluctuation of f -γ (γ = 0.94). Therefore, the source noise noise revealed by AE signals sounds “pink”, supporting the evidences of the cracking process found revealed by AE signals sounds “pink”, supporting the evidences of the cracking process found during during the monitoring campaigns [28,30]. We can say that, in the masonry region monitored by the monitoring campaigns [28,30]. We can say that, in the masonry region monitored by means of AE, means of AE, pink noise fluctuations suggest that the material strength has been locally overcome, as pink noise fluctuations suggest that the material strength has been locally overcome, as in the case of in the case of the laboratory tests described above. the laboratory tests described above. Appl. Sci. 2018, 8, x 9 of 14 Appl. Sci. 2018, 8, 1685 9 of 14 Appl. Sci. 2018, 8, x 9 of 14

γ Figure 11. Log-log plot of AE power spectrum versus frequency for the Asinelli Tower, Bologna: 1/γf Figure 11. Log-log plot of AE power spectrum versus frequency for the Asinelli Tower, Bologna: 1/γf noise,Figure with 11. γ Log= 0.94.-log Pink plot ofnoise AE sourcepower spectrum(Correlated versus phenomena). frequency for the Asinelli Tower, Bologna: 1/f noise,noise, with withγ =γ 0.94.= 0.94. Pink Pink noise noise source source (Correlated(Correlated phenomena). AE technique was successfully applied by the authors, not only to assess historical monument AEAE technique technique was was successfully successfully applied applied byby thethe authors, not not only only to to assess assess historical historical monument monument stability, but also to evaluate the structural supports of mural paintings, and the state of stability,stability, but but also also to evaluate to evaluate the structural the structural supports supports of mural of paintings, mural paintings, and the state and of the conservation state of conservation of the frescos. As a second object of analysis in this work, the results of the monitoring of theconservation frescos. As of athe second frescos. object As a of second analysis object in this of analysis work, the in this results work, of thethe monitoringresults of the of monitoring Chapel XVII of Chapel XVII (also known as “The Transfiguration of Christ on Mount Tabor”) of the Sacred (alsoof known Chapel as XVII “The (also Transfiguration known as “The of ChristTransfiguration on Mount of Tabor”) Christ on of the Mount Sacred Tabor”) Mountain of the of Sacred Varallo isMountain chosenMountain (Figureof ofVarallo Varallo 12). is The ischosen chosen most (Figure (Figure ancient 12). 12). Sacred The The most Mountainmost ancient of Sacred northernSacred Mountain Mountain Italy is of the of northern northern Sacred MountainItaly Italy is isthe the of Varallo,SacredSacred Mountain consisting Mountain of in of Varallo, 45 Varallo, monumental c onsistingconsisting chapels inin 4545 dating monumentalmonumental back to chapelschapels the Renaissance datingdating back back period, to to the the and Renaissance Renaissance it is located atperiod, theperiod, top and ofand ait mountisit islocated located crest, at at amongthe the top top theof of a greena mount mount of thecrest,crest, forests among surrounding thethe greengreen of theof the the City forests forests of Varallo. surrounding surrounding The chapels the the containCityCity of ofVarallo. about Varallo. 800 The The multicoloured chapels chapels contain contain life-size about about terracotta800 800 multicolouredmulticoloured statues, representing lifelife--sizesize terracotta terracotta Life, statue Passionstatues, s,representing andrepresenting Death of ChristLife,Life, Passion [ 31Passion]. and and Death Death of of Christ Christ [31]. [31].

Figure 12. External view of Chapel XVII of the Sacred Mountain of Varallo (Italy). Figure 12. ExternalExternal view of Chapel XVII of the Sacred Mountain of Varallo (Italy). ConcerningConcerning Chapel Chapel XVII XVII structural structural integrity,integrity, a vertical crack crack of of about about 3.00 3.00 m m in in length length and and a a detachmentConcerning of frescosChapel are XVII present, structural both onintegrity, the north a wall,vertical which crack are ofthe about object 3.00 of the m AE in monitoringlength and a detachment of frescos are present, both on the north wall, which are the object of the AE monitoring detachmentcampaign. of AE frescos sensors are are present, employed both to on monitor the north the wall,damage which evolution are the of object the structural of the AE support monitoring of campaign. AE sensors are employed to monitor the damage evolution of the structural support campaign.the decorated AE sensors surfaces are of employedChapel XVII: to fourmonitor are positionedthe damage around evolution the vertical of the crstructuralack, while support two are of of the decorated surfaces of Chapel XVII: four are positioned around the vertical crack, while two thepositioned decorated nearsurfaces the frescosof Chapel detachment. XVII: four For are the positioned sensor pasting around on the valuable vertical decorated crack, while surfaces, two aare are positioned near the frescos detachment. For the sensor pasting on valuable decorated surfaces, positionedsuitable methodology near the frescos is applied detachment. [32]. For the sensor pasting on valuable decorated surfaces, a a suitable methodology is applied [32]. suitable methodology is applied [32]. Appl. Sci. 2018, 8, 1685 10 of 14

Appl. Sci. 2018, 8, x 10 of 14 The equipment used for signal acquisition and processing consists of the aforementioned six The equipment used for signal acquisition and processing consists of the aforementioned six USAMThe units. equipment The signals used acquisition for signal thresholdacquisition was and settled processing at 100 consistsµV. of the aforementioned six USAM units. The signals acquisition threshold was settled at 100 μV. USAMThe monitoringunits. The signals period acquisition of the structural threshold supports was settled of the at chapel 100 μV. began on 28 April 2011 and ended The monitoring period of the structural supports of the chapel began on 28 April 2011 and on 4 June 2011, it lasted about 900 h [32]. As results from the monitoring campaign, the vertical crack ended on 4 June 2011, it lasted about 900 h [32]. As results from the monitoring campaign, the monitored on the North wall of the chapel presents a distribution of cracks on a surface domain, vertical crack monitored on the North wall of the chapel presents a distribution of cracks on a clearly proved by the b-value in the range (0.95, 1.15). Concerning the monitored frescos detachment, surface domain, clearly proved by the b-value in the range (0.95, 1.15). Concerning the monitored the decorated surface tends to evolve towards metastable conditions, and the acquired signals showed frescos detachment, the decorated surface tends to evolve towards metastable conditions, and the highacquired frequency signals characteristics showed high (<400 frequency kHz). Moreover, characteristics a distribution (<400 kHz). of Moreover, microcracks a distribution in the volume of is obtainedmicrocracks for the in analyzedthe volume region is obtained [32]. for the analyzed region [32]. TheThe results results obtained obtained by by the the applicationapplication of the AE sensors sensors are are represen representedted in in a alog log-log-log plot plot of of -γ powerpower spectrum spectrum versus versus frequency frequency (Figure 13),13), with with a afluctuation fluctuation of off-γ (γf =( 0.92).γ = 0.92). Also in Also this in case, this the case, thesource source noise noise revealed revealed by AE by signals AE signals sounds sounds “pink”, “pink”, supporting supporting the evidences the evidences of the cracking of the process cracking processfound found during during the monitori the monitoringng campaigns campaigns [24,28,32]. [24 ,28,32].

Figure 13. Log-log plot of AE power spectrum versus frequency for the Chapel XVII of the Sacred Figure 13. Log-log plot of AE power spectrum versus frequency for the Chapel XVII of the Sacred Mountain of Varallo: 1/f γ noise, with γ = 0.92. Pink noise source (Correlated phenomena). Mountain of Varallo: 1/fγ noise, with γ = 0.92. Pink noise source (Correlated phenomena). Finally, the log-log plot of the AE power spectrum versus frequency for a monitored masonry Finally, the log-log plot of the AE power spectrum versus frequency for a monitored masonry arch [33], belonging to the Racconigi Castle (Figure 14a,b) in Piedmont (Italy) is proposed. This last arch [33], belonging to the Racconigi Castle (Figure 14a,b) in Piedmont (Italy) is proposed. This last exampleexample has has been been presented presented in order in order to demonstrate to demonstrate that athat “white” a “white” response response is obtained is obtained on a masonry on a structuralmasonry element structural in element which the in post-peakwhich the post loading-peak phase loading is farphase from is beingfar from achieved. being achieved. From Figure From 15 , γ it resultsFigure 1/ 15,f itnoise results with 1/fγ noise~ 0.0, with that characterizes γ ~ 0.0, that characterizeswhite sources, white generated sources, by generatedsmall defects by randomsmall diffuseddefects in random the structural diffused volume. in the structural volume.

(a) (b)

FigureFigure 14. 14.External External view view of of the the Racconigi Racconigi CastleCastle (Italy) ( (aa)) and and the the monitored monitored masonry masonry arch arch inside inside the the CastleCastle (b ).(b). Appl. Sci. 2018, 8, 1685 11 of 14 Appl. Sci. 2018, 8, x 11 of 14

Figure 15. Log-log plot of AE power spectrum versus frequency for the monitored masonry arch: 1/f γ Figure 15. Log-log plot of AE power spectrum versus frequency for the monitored masonry arch: 1/fγ noise, with γ ~ 0.0. White noise source (Random phenomena). noise, with γ ~ 0.0. White noise source (Random phenomena). Comparing the results obtained during the laboratory compression tests, and the data collected Comparing the results obtained during the laboratory compression tests, and the data collected from in-situ structural monitoring, we remark a sophistication of the “pink music”, that suggests that from in-situ structural monitoring, we remark a sophistication of the “pink music”, that suggests 1/f noise may have an essential role in self-organized criticality and correlated processes. By applying that 1/f noise may have an essential role in self-organized criticality and correlated processes. By it to damage analysis of materials, the distribution 1/f behaves as a very suitable indicator of critical applying it to damage analysis of materials, the distribution 1/f behaves as a very suitable indicator states. At last, recalling the elastic-softening constitutive law for the structural response [26,34], of critical states. At last, recalling the elastic-softening constitutive law for the structural response when microcracks are organized on preferential fracture surfaces, the damaging structure “produces” [26,34], when microcracks are organized on preferential fracture surfaces, the damaging structure pink noise. “produces” pink noise. It is possible to find out in the previous discussion a parallel with the statistical distribution It is possible to find out in the previous discussion a parallel with the statistical distribution analysis of AE events, the so-called b-value [7–9,35–40]. Therefore, the emitted energy modalities are analysis of AE events, the so-called b-value [7–9,35–40]. Therefore, the emitted energy modalities are identified during the monitoring process by determining the b-value variations. b-value extreme cases identified during the monitoring process by determining the b-value variations. b-value extreme are: the critical conditions b = 1.5, when the energy emission takes place through small defects diffused cases are: the critical conditions b = 1.5, when the energy emission takes place through small defects homogeneously in the structural volume; and b = 1.0, when energy emission takes place throughout diffused homogeneously in the structural volume; and b = 1.0, when energy emission takes place preferential fracture surfaces. throughout preferential fracture surfaces. A diffused damage in the former case is observed (2b = D ' 3.0, i.e., the cracks are distributed A diffused damage in the former case is observed ( 2bD 3.0 , i.e., the cracks are in a fractal domain near a volume [41–43]), whereas in the latter, two-dimensional macrocracks are distributed in a fractal domain near a volume [41–43]), whereas in the latter, two-dimensional formed leading to the separation of the structural element (2b = D ' 2.0, i.e., the cracks are distributed macrocracksin a fractal domain are formed close leading to a surface to the [separation41–43]). Like of the the structuralb-value analysis, element the( 2bD proposed 1/ 2.0f γ noise, i.e., themodel cracks analyzes are distributed the transition in a fractal from the domain diffused close damage to a surface condition [41– to43 that]). Like of imminentthe b-value collapse, analysis, when the γ proposeddamaging 1/ structuresf noise model sound analyzes “pink”. the transition from the diffused damage condition to that of imminent collapse, when damaging structures sound “pink”. 4. Conclusions 4. Conclusions In quasi-brittle materials, like concrete, masonry, and mortar, there is a source of 1/f noise during AcousticIn quasi Emissions-brittle (AE) materials, from fracture like concrete, phenomena. masonry, The and physical mortar, origin there of 1/is f a noise source is still of 1/ anf noise open duringquestion, Acoustic and it is Emissions interesting (AE) to speculate from fracture regarding phenomena. the universality The physical of this origin particular of 1/ noisef noise mechanism. is still an open AEquestion, data detected and it fromis interesting compression to spe testsculate of concrete regarding specimens the universality with different of this slenderness, particular suggest noise mechanism.that 1/f γ noise behaviour is meaningful also in damaging structures. AEIn fact, data before detected the critical from compression state is reached, tests it of appears concrete 0.29 specimens≤ γ ≤ 0:33 with for 1/ differentf γ noise. slenderness, This means suggestthat, when that the 1/fγ damaging noise behaviour is characterized is meaningful by the also formation in damaging of micro-cracks structures. uniformly distributed in the concreteIn fact, before specimen, the critical and the state dominant is reached, fracture it appears surfaces 0.29 have ≤ γ not ≤ yet0:33 been for 1/ generated,fγ noise. This the means energy that,emitted when from the the damaging material is and characterized measured throughby the formation AE signals, of ismicro characterized-cracks uniformly by small distributed variations forin the concrete specimen, and the dominant fracture surfaces have not yet been generated, the energy emitted from the material and measured through AE signals, is characterized by small variations for Appl. Sci. 2018, 8, 1685 12 of 14 every frequency range. The distribution of micro-cracks in concrete sample is therefore random. As a confirmation, the AE sources appear evenly distributed in the damaged body, 2b = D ' 3.0. On the other hand, after the critical state, i.e., in the descending branch of the load-time curve, the preferential fracture surfaces are now generated, and the energy emitted by the damaged material is characterized by significant variations in relation to the frequency: the lower the signal frequency, the higher the emitted energy by the damaging body. In this case, the AE sources appear distributed on preferential fracture surfaces, 2b = D ' 2.0. More precisely, it is interesting to note that, for all the considered cases, during the damage increase, the γ range could be reduced to 0.8 ≤ γ ≤ 0.9 (pink noise). To highlight this observation, AE data series from in-situ Acoustic Emission monitoring are presented: the first one is referred to the Asinelli Tower, the highest construction in Bologna City center, Italy, built in the early XII Century. The second belongs to the investigation of the state of conservation of the mural paintings and their structural supports in the Chapel XVII of the Sacred Mountain of Varallo Renaissance Complex (Italy). AE data obtained from the two damaging structures show a 1/f γ-like power spectrum, with a value of γ close to unity (pink noise), suggesting that 1/f noise may behave as a very suitable indicator of both the critical state and that preceding the collapse. The interpretation of 1/f noise phenomena through the use of the AE technique can be therefore very useful for identifying the transition from the critical conditions of a structure to those that involve an incipient collapse.

Author Contributions: Conceptualization, Methodology, and Draft Preparation: G.L. and F.A.; Validation: A.C. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Milotti, E. 1/f noise: A pedagogical review. arXiv 2002, arXiv:physics/0204033. 2. Mandelbrot, B. Fractals: Form, Chance and Dimension; W. H. Freeman and Company: San Francisco, CA, USA, 1977. 3. Institute for Telecommunication Sciences—National Telecommunications and Information Administration (ITS-NTIA). Federal Standard 1037C: Glossary of Telecommunication Terms; General Services Administration Information Technology Service: Washington, DC, USA, 1996. 4. Dutta, P.; Horn, P.M. Low-frequency fluctuations in solids: 1/f noise. Rev. Mod. Phys. 1981, 53, 497–516. [CrossRef] 5. Bak, P.; Tang, C.; Wiesenfeld, K. Self-organised criticality: An explanation of 1/f noise. Phys. Rev. Lett. 1987, 59, 381–384. [CrossRef][PubMed] 6. Voss, R.F.; Clarke, J. 1/f noise in music and speech. Nature 1975, 258, 317–318. [CrossRef] 7. Carpinteri, A.; Lacidogna, G.; Pugno, N. Richter’s laws at the laboratory scale interpreted by Acoustic Emission. Mag. Concr. Res. 2006, 58, 619–625. [CrossRef] 8. Carpinteri, A.; Lacidogna, G.; Puzzi, S. Prediction of cracking evolution in full scale structures by the b-value analysis and Yule statistics. Phys. Mesomech. 2008, 11, 260–271. [CrossRef] 9. Carpinteri, A.; Lacidogna, G.; Accornero, F.; Mpalaskas, A.C.; Matikas, T.E.; Aggelis, D.G. Influence of damage in the acoustic emission parameters. Cem. Concr. Compos. 2013, 44, 9–16. [CrossRef] 10. Richter, C.F. Elementary Seismolog; W. H. Freeman and Company: San Francisco, CA, USA; Bailey Bros. & Swinfen Ltd.: London, UK, 1958. 11. Carpinteri, A.; Lacidogna, G.; Manuello, A. The b-value analysis for the stability investigation of the ancient Athena Temple in Syracuse. Strain 2011, 47, 243–253. [CrossRef] 12. Aki, A. A Probabilistic Synthesis of Precursory Phenomena. In Earthquake Prediction: An International Review; Simpson, D.W., Richards, P.G., Eds.; American Geophysical Union: Washington, DC, USA, 1981; Volume 4, pp. 566–574. 13. Main, I.G. A modified Griffith criterion for the evolution of damage with a fractal distribution of crack lengths: Application to seismic event rates and b-values. Geophys. J. Int. 1991, 107, 353–362. [CrossRef] Appl. Sci. 2018, 8, 1685 13 of 14

14. Main, I.G. Damage mechanics with long-range interactions: Correlation between the seismic b-value and the two point correlation dimension. Geophys. J. Int. 1992, 111, 531–541. [CrossRef] 15. Main, I.G. A damage mechanics model for power-law creep and earthquake aftershock and foreshock sequences. Geophys. J. Int. 2000, 142, 151–161. [CrossRef] 16. Carpinteri, A.; Lacidogna, G.; Niccolini, G. Fractal analysis of damage detected in concrete structural elements under loading. Chaos Solitons Fractals 2009, 42, 2047–2056. [CrossRef] 17. Carpinteri, A.; Lacidogna, G.; Puzzi, S. From criticality to final collapse: Evolution of the b-value from 1.5 to 1.0. Chaos Solitons Fractals 2009, 41, 843–853. [CrossRef] 18. King, G.C.P. The accommodation of large strains in the upper lithosphere of the earth and other solids by self-similar fault systems: The geometrical origin of b-value. Pure Appl. Geophys. 1983, 121, 761–815. [CrossRef] 19. Hirata, T. A correlation between b-value and the fractal dimension of earthquakes. J. Geophys. Res. 1989, 94, 7507–7514. [CrossRef] 20. Rundle, J.B.; Turcotte, D.L.; Shcherbakov, R.; Klein, W.; Sammis, C. Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems. Rev. Geophys. 2003, 41, 1–30. [CrossRef] 21. Turcotte, D.L. Fractals and Chaos in Geology and Geophysic; Cambridge University Press: New York, NY, USA, 1997. 22. Aggelis, D.G. Wave propagation through engineering materials; Assessment and monitoring of structures through non-destructive techniques. Mater. Struct. 2013, 146, 519–532. [CrossRef] 23. Livitsanos, G.; Shetty, N.; Hündgen, D.; Verstrynge, E.; Wevers, M.; Van Hemelrijck, D.; Aggelis, D.G. Acoustic emission characteristics of fracture modes in masonry materials. Constr. Build. Mater. 2018, 162, 914–922. [CrossRef] 24. Carpinteri, A.; Lacidogna, G.; Invernizzi, S.; Accornero, F. The Sacred Mountain of Varallo in Italy: Seismic risk assessment by Acoustic Emission and structural numerical models. Sci. World J. 2013, 2013, 170291. [CrossRef][PubMed] 25. Carpinteri, A.; Corrado, M.; Lacidogna, G. Heterogeneous materials in compression: Correlations between absorbed, released and acoustic emission energies. Eng. Fail. Anal. 2013, 33, 236–250. [CrossRef] 26. Carpinteri, A.; Accornero, F. Multiple snap-back instabilities in progressive microcracking coalescence. Eng. Fract. Mech. 2018, 187, 272–281. [CrossRef] 27. RILEM Technical Committee (Masayasu Ohtsu). Recommendation of RILEM TC212-ACD: Acoustic Emission and related NDE techniques for crack detection and damage evaluation in concrete: Measurement method for acoustic emission signals in concrete. Mater. Struct. 2010, 43, 1177–1181. [CrossRef] 28. Lacidogna, G.; Accornero, F.; Carpinteri, A. Masonry structures. In Innovative AE and NDT Techniques for On-Site Measurement of Concrete and Masonry Structures (RILEM); Ohtsu, M., Ed.; Springer: Heidelberg, Germany, 2016; Chapter 3, pp. 27–46. 29. Lacidogna, G.; Manuello, A.; Niccolini, G.; Accornero, F.; Carpinteri, A. Acoustic Emission Wireless Monitoring of Structures. In Acoustic Emission and Related Non-Destructive Evaluation Techniques in the Fracture Mechanics of Concrete; Ohtsu, M., Ed.; Woodhead Publishing: Cambridge, UK, 2015; Chapter 2, pp. 15–40. 30. Carpinteri, A.; Lacidogna, G.; Manuello, A.; Niccolini, G. A study on the structural stability of the Asinelli Tower in Bologna. Struct. Control Health Monit. 2016, 23, 659–667. [CrossRef] 31. Niccolini, G.; Borla, O.; Accornero, F.; Lacidogna, G.; Carpinteri, A. Scaling in damage by electrical resistance measurements: An application to the terracotta statues of the Sacred Mountain of Varallo Renaissance Complex (Italy). Rend. Lincei—Sci. Fis. Nat. 2014, 26, 203–209. [CrossRef] 32. Accornero, F.; Invernizzi, S.; Lacidogna, G.; Carpinteri, A. The Sacred Mountain of Varallo renaissance complex in Italy: Damage analysis of decorated surfaces and structural supports. In Acoustic, Electromagnetic, Neutron Emissions from Fracture and Earthquakes; Carpinteri, A., Lacidogna, G., Manuello, A., Eds.; Springer: Heidelberg, Germany, 2015; Chapter 17, pp. 249–264. 33. Niccolini, G.; Manuello, A.; Marchis, E.; Carpinteri, A. Signal frequency distribution and natural-time analyses from acoustic emission monitoring of an arched structure in the Castle of Racconigi. Nat. Hazards Earth Syst. Sci. 2017, 17, 1025–1032. [CrossRef] 34. Lacidogna, G.; Accornero, F. Elastic, plastic, fracture analysis of masonry arches: A multi-span bridge case study. Curved Layer Struct. 2018, 5, 1–9. [CrossRef] Appl. Sci. 2018, 8, 1685 14 of 14

35. Kurz, J.H.; Finck, F.; Grosse, C.U.; Reinhardt, H.W. Stress drop and stress redistribution in concrete quantified over time by the b-value analysis. Struct. Health Monit. 2006, 5, 69–81. [CrossRef] 36. Shiotani, T.; Fujii, K.; Aoki, T.; Amou, K. Evaluation of Progressive Failure using AE Sources and Improved b-value on Slope Model Tests. Prog. Acoust. Emiss. 1994, 7, 529–534. 37. Scholz, C.H. The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes. Bull. Seismol. Soc. Am. 1968, 58, 399–415. 38. Colombo, S.; Main, I.G.; Forde, M.C. Assessing damage of reinforced concrete beam using “b-value” analysis of acoustic emission signals. J. Mater. Civ. Eng. ASCE 2003, 15, 280–286. [CrossRef] 39. Rao, M.V.M.S.; Prasanna Lakshmi, K.J. Analysis of b-value and improved b-value of acoustic emissions accompanying rock fracture. Curr. Sci. 2005, 89, 1577–1582. 40. Lockner, D.A.; Byerlee, J.D.; Kuksenko, V.; Ponomarev, A.; Sidorin, A. Quasi static fault growth and shear fracture energy in granite. Nature 1991, 350, 39–42. [CrossRef] 41. Lacidogna, G.; Accornero, F.; Carpinteri, A. Influence of snap-back instabilities on Acoustic Emission damage monitoring. Eng. Fract. Mech. 2018.[CrossRef] 42. Carpinteri, A.; Corrado, M.; Lacidogna, G. Three different approaches for damage domain characterization in disordered materials: Fractal energy density, b-value statistics, renormalization group theory. Mech. Mater. 2012, 53, 15–28. [CrossRef] 43. Carpinteri, A.; Lacidogna, G.; Niccolini, G.; Puzzi, S. Morphological fractal dimension versus power-law exponent in the scaling of damaged media. Int. J. Damage Mech. 2009, 18, 259–282. [CrossRef]

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).