Int J Fract (2020) 226:233–242https://doi.org/10.1007/s10704-020-00490-y

ORIGINAL PAPER

Non-monotonic precursory signals to multi-scalecatastrophic failures

Hu Wang · Sheng-Wang Hao · Derek Elsworth

Received: 20 July 2020 / Accepted: 26 September 2020 / Published online: 22 October 2020© Springer Nature B.V. 2020

Abstract Identifying precursory trends in acous-tic/seismic observations allows the forewarning/predi-ction of catastrophic events. However, rupturing acrossmultiple scales leaves it unclear whether features ofsmall events are applicable predictors of the largerensemble final collapse. To resolve this issue, wepresent a multiscale heterogeneousmodel that straight-forwardly characterizes the duration and mechanismof multiscale catastrophic failures. Our results identifyfour distinct classes of failure including random singlebreaks, small catastrophic failure (SCF) events, largecatastrophic failure (LCF) events that consist of subor-dinate SCF and randombreak events, and a culminatingmacroscopic catastrophic failure (MCF) event result-ing from the coalescence of subordinate LCF events.Only the local response quantities, recorded at their cor-responding position, show an accelerating precursorytrend to an SCF event. LCF events can appear in stagesboth before and after the maximum load in the sys-tem. Our findings highlight that although cumulativeLCF event and deformation rates for the entire systemalways exhibit singular accelerating precursors asMCF

H. Wang · S.-W. Hao (B)School of Civil Engineering and Mechanics, HebeiProvincial Key Laboratory of Green Construction andIntelligent Maintenance of Civil Engineering, YanshanUniversity, Qinhuangdao, Chinae-mail: hsw@ysu.edu.cn

D. ElsworthEnergy and Mineral Engineering, Geosciences, G3 Center, andEMSEnergy Institute, Pennsylvania State University, UniversityPark, PA, USA

is approached, this is not true at all individual eventpoints. This may explain why no clearly acceleratingprecursor is observed before some catastrophic events.Thus, these results suggest amethodology for recogniz-ing and distinguishing effective precursory informationfrom monitoring signals across scales and in eliminat-ing false predictions.

Keywords Catastrophic failure · Multi-scale events ·Precursory signals · Non-monotonic · Prediction offailure · Heterogeneous materials

1 Introduction

Catastrophic failure in heterogeneous materials hasattracted substantial interest (Kilburn and Petley 2003;Main and Naylor 2012; Heap et al. 2011; Hao et al.2014; Sornette 2002) due to its relevance to a broadarray of engineered and natural systems. These includephenomena spanning structures in civil, mechanical,marine, aeronautics and astronautics engineering andprocesses in natural hazards including earthquakes,volcanic eruptions, landslides and avalanches. Hetero-geneities at different scales result in a broad spectrumof complex failure behaviors (Sornette 2002; Kilburn2003; Zapperi et al. 1997; Vasseur et al. 2015, 2017;Kadar and Kun 2019) making prediction of failuredifficult. Conversely, this high degrees of heterogene-ity and resulting disorder in materials may produce acumulative trend in damage as failure is approached

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with distinct features—thereby offering a unique signa-ture of precursory phenomena (Sornette 2002; Kilburn2003; Vasseur et al. 2017; Kadar and Kun 2019; Mogi1995; Hao et al. 2017a, b). Evolution towards a macro-scopic catastrophic eventmay comprisemultiple physi-cal processes at different scales and their possible inter-dependences. Therefore, faithful representation of themulti-scale nature of failure in heterogeneous materi-als is crucial for our understanding of rupture and infailure prediction (Kilburn 2003; Zapperi et al. 1997;Vasseur et al. 2015). However, the specific connectionof a large culminating event to the contributory smallerscale events remains unclear. This creates a challengein identifying representative signals that are precursoryto the impending catastrophic event.

Analyzing the precursors to catastrophic failure isa long-standing problem—although approaches havebeen codified and widely used to predict materialfailures. Typically, acceleration in “response quanti-ties”, such as accumulative seismic events, acousticemissions or deformations, as catastrophic failure isapproached, are used in the field monitoring of vol-canic eruptions (Cornelius andVoight 1995;Voight andCornelius 1991; Kilburn and Voight 1998), landslides(Kilburn 2003; Petley et al. 2005), earthquakes and inlaboratory experiments (Heap et al. 2011; Hao et al.2014; Nechad et al. 2005; Hao et al. 2017a, b; Voight1989). This accelerating process can be described byVoight’s relation (Voight 1988, 1989; Cornelius andScott 1993) and presents a power law relationship(Voight 1988, 1989; Kilburn and Voight 1998; Petleyet al. 2005; Hao et al. 2014; Nechad et al. 2005; Haoet al. 2017a, b; Cornelius and Scott 1993; Main 1999,2000; Turcotte et al. 2003; Bell et al. 2011; Boué et al.2015; Zhou et al. 2018) with respect to time-to-failure.The emergence of Voight’s relation may be explainedby applying statistical mechanics to rock fracture (Kil-burn 2003, 2012) and to extend analyses to deformationunder a changing stress regime (Kilburn 2012). Sucha power law relation can be expressed as the relativechange in response quantities (Kilburn 2012; Hao et al.2017a, b) with respect to the controlling load variableof stress (Kilburn 2012; Hao et al. 2016) or boundarydisplacement (Hao et al. 2013, 2017; Xue et al. 2018).But this form of accelerating precursor is not alwaysobserved before some catastrophic failure events. Thecause for the absence of this accelerating precursorremains unclear.

Macroscopic fractures grow from the coalescence ofmicro-cracks. Cracks at all scales propagate unstablywhere the energy released from the surrounding massexceeds that required for nucleation and propagation(Griffith 1921). Elastic energy release is a fundamen-tal mechanism contributing to dynamic and unstablefailure with short-warning-times such as the failure ofstructures in rock (Salamon 1970; Jaeger et al. 2007)and abrupt failure of specimens tested in the laboratory(Hao et al. 2013, 2014; Xue et al. 2018). This intrinsicmechanism results in failures cascading over a spec-trum of scales when the local energy release rate drivesthe response. However, the cross-scale relationships ofthese catastrophic events remain unclear.

Acoustic emissions (AE) or seismic events impli-cated in macroscopic failure evolve across a spectrumof length- and time-scales. Recognizing their multi-scale evolution and precursory signatures is essentialfor improving time-to-failure warnings (Kilburn et al.2018). Models must anticipate potential oscillations inseismic event rates and identify that local peaks in localevent rate (i.e., the inverse-rate minima), rather than allseismic events, play a key indicator in approaching fail-ure (Kilburn 2003). This global predictive method hasbeen validated against field (Kilburn 2003) and lab-oratory observations (Lavallée et al. 2008). However,an underlying unresolved issue is in determining whatscale of smaller events are directly related to largeror culminating catastrophic events - i.e. what scale ofsmall events are suitably precursory to a larger event.

Based on the intrinsic mechanism of energy releasedriving catastrophic failure, we develop a multi-scalemodel to address this issue of event size consistency.The model consists of elastic springs and damageablelinks in a variety of configurations. Thismodel providesdirect illustration of mechanisms and signatures ofmulti-scale catastrophic failures. Energy release fromthe elastic springswhen far-from-overall-failure resultsin small catastrophic failure in some distributed meso-elements. Energy release from the elastic springs at twoscales drives a large catastrophic event that involvesrandom distributed breaks and smaller catastrophicevents. Macroscopic catastrophic failure occurs whensuch cumulative events coalesce in space and time. Ourfindings show that event size grows as the system accel-erates to macroscopic catastrophic failure, but localcumulative counts do not necessarily show this criti-cal trend. The smaller breaks in a local meso-element

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Non-monotonic precursory signals to multi-scale catastrophic failures 235

Fig. 1 Multi-scale systemconsisting of a primaryelastic spring with astiffness of ke and asub-system in series. Thesub-system comprisesmultiple meso-elements inparallel. Each meso-elementcombines an elastic springwith a stiffness of kem inseries with a fiber bundle ofmany parallel and breakablefibers

only play a precursory signal to local small catastrophicfailure.

2 Multi-scale model and numerical calculations

To demonstrate the occurrence of multi-scale catas-trophic failures driven by energy release, we focuson a system consisting of a linear elastic spring anda sub-system in series as shown in Fig. 1. The sub-system comprises M meso-elements in parallel. Eachmeso-element consists of a linear elastic spring andNm damageable links in series. The damageable linksare elastic-brittle fiber bundles and represent a classof simple models widely used (Peirce 1926; Sornette1989; Duxbury et al. 1995; Hidalgo et al. 2002; Prad-han et al. 2010; Kun et al. 2003; Moreno et al. 2000) toexplain evolving failure mechanisms in heterogeneousmaterials— and proven effective in the study of fail-ure process of disordered materials (Hao et al. 2016,Hao et al. 2017a, Hao et al. 2017b; Hidalgo et al. 2002;Pradhan et al. 2010; Kun et al. 2003; Moreno et al.2000). Specifically in this work, we build a multi-scalesystem with heterogeneities at different scales repre-senting heterogeneous elastic environments at differ-ent spatial positions and at different scales. The bound-ary displacement U of the macro-system is the sumof the uniform deformation um of the sub-system andthe deformation of the primary elastic spring with astiffness of ke (Fig. 1).

A global load-sharing criterion is chosen for theredistribution of load following a break in the meso-element fibers. All the fibers are linearly elastic andassumed to have the same stiffness until they break.A fiber breaks when it reaches its strength— subse-quently it carries no load. The surviving fibers in ameso-element equally share the force released by thebrokenfibers.Allmeso-elements share the load accord-ing to their stiffness ratio with respect to the total stiff-ness of the sub-system. A meso-element no longer car-ries load when it has no surviving fibers.

The stiffness of a single fiber is assumed to beunity. Then, the normalized force ( f )-deformation (u)

response of an unbroken fiber has the relationshipf = u. The nominal force in each meso-elementis then fm = f (Nm − Nbm)/Nm where Nbm is thenumber of broken fibers in the meso-element. As a

consequence, the nominal force is f0 =M∑

m=1fm/M

on the macroscopic systemwith M meso-elements and

the total number of broken fibers Nb =M∑

m=1Nbm is the

sum of those in all meso-elements.The strength of fibers in each meso-element is

assumed to follow the Weibull distribution P( fth) =1 − exp[−( fth/η)θ ]. In order to describe the hetero-geneity among meso-elements, parameters η and θ aredifferent amongmeso-elements and follow theWeibulldistribution P(η) = 1−exp(−ηγ ) for η and a uniformdistribution ranging from 2.0 to 6.0 for θ .

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N

N

u

uu

NN

u

N

N

u

u

U UU

U

f

f

f f

A

B

C

D

E

F

G

H I

Fig. 2 Evolution of catastrophic failure across different scales.a Evolution of breakage of fibers. b um (u) vs U . c Evolutionof force vs U . Evolution in the response of three representativemeso-elements is shown to illustrate small-catastrophic-failure(SCF) occurring in the meso-elements. d Cumulative numberof events (Ne, NLCF) vs U . e, f, g, h, i). Zoom-in of LCF

events showing their relationshipwith, anddifferences from,SCFevents. Gray solid diamonds and empty triangles in subplots E, FandG represent typical evolution (Nbm and u versusU ) curves oftwo meso-elements, respectively, where SCF events occur Thetotal number of meso-elements in the simulations is M =200and the number of fibers in each meso-element is Nm =100

In order to represent multi-scale accelerating fail-ures, Monte Carlo simulations of the failure processare performed. In these calculations, the stiffness ofthe primary elastic spring connected to the sub-systemis set to 1.4 Nm since every fiber has a unit stiffness, andthe stiffnesses of elastic springs in the meso-elementsfollow a uniform distribution ranging from 0.4 Nm to

1.4 Nm. The load process is driven as the boundarydisplacement U is monotonically increased at a suf-ficiently slow rate that the breaking fibers increase toproduce a smallest positive increment ofU at each cal-culation step.

The calculation sequence proceeds as follows: Letfibers break one by one, i.e. every time let only a single

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Non-monotonic precursory signals to multi-scale catastrophic failures 237

fiber break that requires the minimum increment of theboundary displacement. Thus, when one fiber breaks ina meso-element, all other meso-elements deform elas-tically because no additional fibers break within them.When a fiber with a strength of fim (uim representsthe corresponding deformation of this fiber) in a meso-element breaks, then the load supported by this meso-element becomes fim(Nm − Nbm) and its deformationis um = fim(Nm−Nbm)/kem+uim . Then the deforma-tions of othermeso-elements are identically determinedas um based on their geometric relationship of parallelconnection shown in Fig. 1, and their forces can be cal-culated through the elastic force-deformation relation-ship. The nominal force f0 and the boundary displace-ment U on the system are correspondingly calculated.After some meso-elements transit to their post-peakforce, the decrease in the load supported by a meso-element due to an internal fiber-break may result ina decrease in the resultant nominal force on the sys-tem, and thus the primary elastic spring unloads andreleases its deformation energy. Thus, this fiber breakcould result in a negative increment in the boundarydisplacement. Subsequently, we will find the next fiberthat requires the minimum increment (may be nega-tive or positive) of the boundary displacement and let itbreak. We repeat this process until the resultant incre-ment of the boundary deformation in this step �U islarger than zero. This progressive breaking process offibers builds towards an intermediate catastrophic fail-ure event—at which this load step is complete. Thiscalculation cycle is repeated until the system fails in itsentirety. This method follows the failure process andis used to understand signatures of discernable failuresignals.

3 Results of multiscale catastrophic failure events

Figure 2 shows calculated results for the response quan-tities with respect to the control variable of system dis-placement U . Response quantities include: the defor-mation of the sub-system um, number of broken fibersNb, breakage event counts NE, and others. With amonotonic increase in system displacement, U , fourkinds of break events spontaneously appear at differ-ent scales. These include the random breakage of asingle fiber, and what we classify as small catastrophicfailure (SCF) events (occurring in meso-elements), alarge catastrophic failure (LCF) events (occurring in the

system) and macroscopic catastrophic failure (MCF)events (leading to the overall failure of the system). Theenergy release of the elastic spring in a meso-systeminduces an SCF event (Fig. 2a), and correspondingly ajump in deformation u (Fig. 2b) and force fm (Fig. 2c)in the fiber bundle.

Every LCF event combines an SCF event in onemeso-element and some distributed breaking of fibersin other meso-elements that is driven by energy releaseof the primary spring connected to the sub-system.Thus, an LCF event could result in the breaking ofmore fibers than the SCF event (see Fig. 2e). An exam-ple shown in Fig. 2f and g illustrates that an SCF eventleads to a large jump (�u) in deformation of the fiberbundle but a much lower increment (�um) in deforma-tion of a meso-element. An LCF event only induces asmall fluctuation in the global trend of the force dis-placement curve (Fig. 2c, h and i).

LCF events show a clearly cumulative trend withincreasing U , i.e. in approaching an MCF event thefrequency of LCF events increases (Fig. 2a–c). Thistrend can be further directly observed in the curve ofNLCF vs U illustrated in Fig. 2d.

The normalized break rate �Nb/�U of fibers(Fig. 3a), and deformation �um/�U , and event rates�NE/�U (Fig. 3c) do not show accelerating trendswhen they approaches the MCF event. NE representsthe cumulative number of all events, including bothLCF events and random breaking of fibers between twoLCF events with a monotonic increase inU . However,the LCF event rate (�NLCF/�U ) shows a clear accel-erating trend as theMCF event is approached (Fig. 3d).The deformation rate (�uLCF/�U ) recorded at theoccurrence of the LCF events also presents a similaraccelerating trend (Fig. 3e).

The linear parts of the log-log plots (Fig. 4) of theLCF event rate (�NLCF/�U ) and deformation rate(�uLCF/�U ) relative to progressive deformation (UF-U ) show that these two rates can be well describedusing a power law relationship

�R/�U ∼ (UF −U )−β (1)

where R represents the cumulative response quanti-ties of NLCF and uLCF. The fitted exponents of β are∼0.5 for�uLCF/�U and∼0.55 for�NLCF/�U . Theseimply that the accelerating precursory trend to anMCFevent could be hidden by the small events occurringbetween LCF events if all events are indeed recorded.

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Fig. 3 Rate of change inresponse quantities formonotonically increasingU . a Break-rate of fibers, bDeformation-rate recordedat each event point. cEvent-rate. d LCFevent-rate. eDeformation-rate recordedonly at LCF event points

ΔN

ΔU

Δum

Δ UΔN

ΔUΔ

ΔU

ΔΔU

U

U

A

B

C

D

E

The initiation of a volcanic eruption may be viewedas prompted by the movement of magma to the sur-face.This occurs bydeveloping afluid transmissive net-work by extending then linking pre-existing fracturesprogressively along the axial cylinder representing theevolving magmatic conduit – and opening a pathwayfor magma to reach the surface. Each fracture itself hasa process zone containing smaller cracks. A cycle ofgrowth and coalescence among the smaller cracks in

the process zone of a fracture leads to oscillations inmonitored seismic event rate with time (Kilburn 2003).Thus, the rate of increase in peak event rate (ratherthan all seismic events) is suggested as a key indicatorof the approach to eruption (Kilburn 2003). Similarly,rheological experiments with continuous microseismicmonitoring have shown that peaks in the event rate bestpredict the path to failure (Lavallée et al. 2008).

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Non-monotonic precursory signals to multi-scale catastrophic failures 239

U

U U

uU

Fig. 4 Power law singularity trends for cumulative LCF eventsas MCF is approached

4 Mechanism of multi-scale catastrophic failuresand accelerating singularity precursors

The geometric condition of a meso-element gives thatthe increment of displacement is the sum of the defor-mation increment of the spring �uem and that of thefiber bundle �u in this meso-element, as,

�um = �uem + �u (2)

When this meso-element transits to its post-peak forcestage, the spring within it unloads and recovers itsdeformation, and thus �uem becomes negative. Whenthe deformation recovery of the spring is sufficient tocompensate for the necessary deformation �u drivingthe breaking of fibers in ameso-element, the total defor-mation um of this meso-element decreases.

Figure 5 illustrates the analytical results of the com-plete curves of u versus um and fm versus um (u)

for an ideal continuous case with an infinite numberof fibers where the deformation u of the fiber bundleis monotonically increased. Alternatively, if the load-controlling variable is changed to the total deformationum of this meso-element, the monotonically continu-ous increase of um will induce a discontinuous jump ofu and the number of breaking fibers along the tangentdirection (vertical direction) frompointC toDas shownin Fig. 5a, i.e., a catastrophic failure (an SCF event).This results in a discontinuous jump in the curve of fmversus u along the tangent line with a slope of –kem(Fig. 5b). At the point of incipient catastrophic failure,an infinitesimal increment of um results in a finite incre-ment of u and a serial breaking of fibers fromC toD, i.e.(du/dum)c → ∞ (Fig. 5a)—and (d fm /dum)c → ∞for the fm versus um curve (Fig. 5b). Thus, the mecha-nism of catastrophic failure determines the acceleratingsingularity precursor.

Similarly, the geometric condition of the macro-scopic system gives that

�U = �um + �ue (3)

where�ue is the deformation increment of the primaryelastic spring connected to the sub-system. When therequired deformation increment �um resulting in anSCF event is larger than the deformation recovery of theprimary elastic spring, the macro-system is stable andonly an SCF event appears in ameso-element. Then theglobal force-deformation ( f0 vsU ) curve is smooth andcontinuous as shown in Fig. 2h but a small oscillation

um

N uu u

u N

bm

f f u

um u

k

f uA B

Fig. 5 Analytical results of complete curves for an ideal continuous case when m=2, k =0.3. a u (Nbm) versus um, B) fm versus um(u). This illustrates the mechanism of catastrophic failure that hosts the precursor accelerating to a singularity

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k Nk N

f

U

k Nk N

U

k Nk N

f

U U

A B

Fig. 6 Nominal force vs. boundary displacement for the primaryelastic spring with different stiffness ratios. aComplete curves. bZoom-in to showLCF events inducing discontinuity in the force-

displacement curves. Scatter yellow circles in symbols representthe response quantities recorded at LCF events

occurs in the um vsU curve. For this case, an LCF eventonly includes an SCF event. Otherwise, an incrementof �U will lead to the distributed random breaking offibers in other meso-elements in addition to an SCFevent (e.g. the case shown in Fig. 2e and f). This LCFevent leads to a discontinuity of the evolving form off0 versus U (Fig. 2i). This LCF event is a combinedresult of energy release from both the spring connectedto sub-system and the spring within a meso-element.LCF events can appear both before and after the pointof maximum load (Fig. 2i).

Based on statistical theory describing the evolu-tion of mesoscopic defects, the damage-failure transi-tion is considered (Naimark 2004, 2017; Naimark andUvarov 2004) as a specific type of critical phenom-ena of structural scaling transitions that could resultin kind of catastrophic events mentioned in this paper.In their model, the transition is related to collectivemodes of damage localization that is responsible forthe nonlinear properties of the free energy releaserate under dynamic and shock wave loading (Naimark2004, 2017; Naimark and Uvarov 2004).

When energy release from the primary elastic springis sufficient to induce more than a single LCF event,it leads to the coalescence of LCF events and thusan MCF event. The stiffness of the elastic springdetermines its deformation recovery rate and energyrelease rate. Figure 6 shows results when the stiffnessratio of the primary elastic spring has other values.When it is larger than the maximum negative valueof the slope of the force–deformation curve of the

sub-system recorded at LCF events, no MCF eventappears (Fig. 6). The cumulative number of LCF events(Figs. 2d and 3d) and deformation (Figs. 2c and 3e)recorded at LCF events shows a precursory trend asit evolves to an MCF event. For an ideal continuoussystem with an infinite number of meso-elements, therate �uLCF/�U (or �NLCF/�U ) continuously andsmoothly evolves to a singularity as it approaches theMCF point because an infinitesimal increment of Uwill result in a series of LCF events. Thus, the rate�uLCF/�U (or �NLCF/�U ) exhibits an acceleratingtrend as it approaches the singular point.

It can be seen that at the catastrophic failure pointdR/dλ tends to a singularity and thus dλ/dR tendsto zero. In this, R represents the “response quantity”directly related with the catastrophic event at the cor-responding scale and λ represents the “load control-ling variable”, such as the boundary displacement Uin the present model and λF is its value at the catas-trophic point. In the vicinity of the point of catastrophicfailure, if dλ/dR can be approximately expressed asan n-th order infinitesimal of (λF-λ), i.e. the limitof dλ/dR/(λF-λ)n tends to a finite quantity C , thendλ/dR = C(λF-λ)n . Thus, dR/dλ represents a powerlaw accelerating trend

dR/dλ = C−1(λF − λ)−β (4)

with λ approaching to λF. Where β =1/n.

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Non-monotonic precursory signals to multi-scale catastrophic failures 241

5 Conclusions

Our multiscale model successfully reproduces catas-trophic events at different scales by representing theunderlying physics of multiscale catastrophic events.The proposedmethodology gives accurate results of thefailure process driven only by a monotonically increas-ing boundary displacement. Small catastrophic (SCF)events appear both before and after the maximum loadpoint on the macroscopic system. An SCF event occur-ring at small scale is driven by the local energy releasefrom the corresponding elastic environment. This smallevent induces a clear jump in the local (andmeasurable)response driven by the progressive break of fibers in themeso-element, but leads to a much smaller (or immea-surable) fluctuation in the trend of the global response.Thus a small catastrophic failure (SCF) event may beobserved locally but may not be available as a record-able precursory signal at the scale of the full system.

At the point of macroscopic catastrophic failure(MCF), the energy release from the system is sufficientto drive the failure process. Thus, a monotonic increasein the controlling variable leads to a disproportionateincrease in the response quantities due to the coales-cence of individual breakage events. For an ideal con-tinuous system with an infinite number of elements, aninfinitesimal increment of the controlling variable willresult in a finite increment of the response quantities.This mechanism hosts a singularity precursor of therelative increase of response quantities with respect tothe controlling variable.

In a multiscale system, failure is able to evolveacross various length-scales. An LCF event can bedrivenbycombining energy release fromelastic springsat different scales. The coalescence of LCF eventsforms a macroscopic catastrophic failure (MCF) event.Thus, only the cumulative rates of LCF events, but notall events, exhibit a power law precursory trend accel-erating to a singularity as it approaches its catastrophicpoint. This is because the much smaller events do notpresent a critical trend as the system approaches thecatastrophic point – and a precursory signature is there-fore not visible. This implies that it is crucial to recog-nize and distinguish effective precursors frommonitor-ing signals that may be relevant to different scales. Thisis a key for the robust prediction of these events, andmay be a possible cause that no obvious acceleratingprecursory signal is observed in advance of some largecatastrophic events.

Acknowledgements This work is supported by National Nat-ural Science Foundation of China (Grant No. 11672258) andNatural Science Foundation of Hebei Province (Grant No.D2020203001).

Compliance with ethical standards

Conflict of interest The authors declare no competing financialinterests.

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