a physics-based model of nonlinear elasticity of rocks for

A physics-based model of nonlinear elasticity of rocks

for reversible damage, slow dynamics, hysteresis and

dynamic acoustoelastic testing (DAET)

Christoph Sens-Schonfelder

(1), Roel Snieder

(2), and Xun Li

(2)

(1) GFZ German Research Centre for Geosciences, Potsdam, Germany

email [email protected]

(2) Center for Wave Phenomena and Dept. of Geophysics, Colorado School of Mines, Golden Co 80401

ABSTRACTCo-seismic changes of the seismic velocity and the subsequent recovery are thefield equivalents to modulus changes in resonant bar experiments or dynamicacoustoelastic testing (DAET) in the laboratory. We present a physics-basedmathematical model for the nonlinear mesoscopic elasiticiy of rocks that de-pends on the instantaneous strain and the strain rate. The model is based onthe two competing processes of damage and healing that alter the sti↵ness ofthe material. Healing constantly increases sti↵ness at a broad range of timescales while damage lowers the sti↵ness proportional to strain rate. The inter-play of these two processes can reproduce a variety of observations including(A) decrease of the mean modulus due to dynamic strain, (B) slow dynamicrecovery, (C) shape of the nonlinear signatures observed in DAET, (D) orienta-tion of the DAET signatures and (E) frequency dependence of the nonlinearity.The damage and healing behavior captured by the mathematical model can bereproduced with a simple model for the creation and destruction of bonds insmall cracks. There are no constraints on the type of the healing process, but anessential feature is that healing sti↵ens the contacts that are currently in placeas known from contact aging and thereby reinforces the present strain state.

Key words: nonlinear elasticity, hysteresis, damage, healing

1 INTRODUCTION

The velocity of seismic wave propagation in elastic me-dia is not a simple material constant. It depends onexternal conditions like the ambient stress level (Mur-naghan, 1944; Birch, 1947; Murnaghan, 1951; Landauet al., 1960). In heterogeneous materials such as rocks,concrete or granular materials such as unconsolidatedsediments the stress dependence is elevated by the sti↵-ness contrast between competent grains and weak con-tacts (grain boundaries, cracks, et.) that lead to strainaccumulation. Such materials exhibit nonlinear meso-scopic elasticity (NME, Guyer and Johnson, 1999) in-dicating that the nonlinearity originates on a spatialscale between the atomic scale of crystal latices andthe macroscopic scale at which the nonlinearity is ob-served. Apart from the elevated stress sensitivity (Guyer

and Johnson, 2009), NME materials exhibit temporalchanges of the elastic properties in response to dynamicdeformation. Upon dynamic excitation by high ampli-tude elastic waves the NME materials are damaged andtheir elastic modulus decreases (Johnson and Sutin,2005; Zaitsev et al., 2014). When excitation terminatesthe modulus recovers in a healing process over a timespan that can be much longer than the duration of theexcitation (TenCate et al., 2000; TenCate, 2011). Weuse the terms damage and healing here to refer to pro-cesses that decrease or increase sti↵ness, respectively,anywhere in the bulk material no matter whether theseprocesses are reversible or not. This is di↵erent from theuse of damage and healing in damage mechanics or rockphysics (e.g. Gratier , 2011) and we use these terms in

2 C. Sens-Schonfelder, R. Snieder, and X. Li

a general way also for processes with small activationenergies.

This damage and healing behavior is not only ob-served in laboratory experiments with ultrasound, it isalso observed in seismological field measurements (Sens-Schonfelder and Wegler , 2011) indicating that NME isnot a peculiarity of high frequency acoustic lab experi-ments but rather a general property of Earth’s materi-als with consequences for seismology. A physical under-standing of the involved mechanism has been hinderedso far by the complexity of the involved processes. Inter-pretations of field observations of time-dependent seis-mic velocities thus rely on di↵erent assumptions aboutthe causative processes. Supported by observation froma novel type of laboratory experiments called DynamicAcoustoelastic Testing (DAET) we introduce a modelthat allows to explain damage due to dynamic strain,slow dynamic recovery and most of the observations inDAET experiments with NMEmaterials. Our treatmento↵ers a simple formulation to model the modulus vari-ations of multiple excitations as commonly observed inseismology.

We begin with an overview of earlier work in sec-tion 2 in which we describe seismological observations,relevant laboratory experiments with special focus onDAET and existing models for reversible changes ofelastic properties. Our mathematical model for DEATobservations is presented in section 3 and results areshown in section 4. In section 5 we discuss the physicalmeaning of the components in the mathematical modeland present a physical model for the dynamic weakeningbefore we conclude in section 6.

2 OVERVIEW OF EARLIER WORK

2.1 Seismological Observations of SeismicVelocity Changes

Over the last decade the seismological observation oftemporal variations of material properties has madeconsiderable progress due to the use of ambient seis-mic vibrations (Sens-Schonfelder and Wegler , 2011).Changes of the seismic velocity as the most commonlymonitored property have been observed in a variety ofenvironments and in connection with di↵erent processes.At volcanoes, observations of velocity changes related tovolcanic activity have been linked to the induced staticstress changes and the resulting opening or closing ofcracks (Brenguier et al., 2008a; Sens-Schonfelder et al.,2014; Budi-Santoso and Lesage, 2016; Donaldson et al.,2017). Depending on the depth of the pressure sourceand the topography, the velocity changes can be pos-itively or negatively correlated with pressure changesand velocity increase and decrease in response to thestatic stress changes is observed.

Clarke et al. (2013) speculate that large-scale flankmovement in response to magma injection can reduce

the seismic velocity. A clear connection of seismic ve-locity reduction with the mobilization of a slope wasdocumented by Mainsant et al. (2012) with continuousmonitoring of a clay landslide. On a completely di↵erentscale Rivet et al. (2011) found a connection between ma-terial deformation and seismic velocity decrease duringa slow slip event in Mexico.

Velocity changes are also observed in response todynamic strain perturbations caused for example by thepassage of large amplitude seismic waves. Such changeslead to a reduction of the seismic velocity. They are mostprominent in the fault area and have been observed us-ing repeating earthquakes (Poupinet et al., 1984; Scha↵and Beroza, 2004; Rubinstein and Beroza, 2005), arti-ficial sources (Nishimura et al., 2000; Vidale and Li ,2003). Wu et al. (2009a,b) identified the co-seismic ve-locity decrease as a change of the peak in the spectralratio between di↵erent seismic stations. Even thoughmost of these observations are reported from stationsclose to the fault, they do not imply that processes areinvolved which occur on the fault or in its direct vicin-ity. Deconvolution between borehole and surface sta-tions was also used to detect time-lapse velocity changesand showed that e↵ects are strongest close to the sur-face (Sawazaki et al., 2009; Sawazaki and Snieder , 2013;Nakata and Snieder , 2011). The best temporal resolu-tion is obtained using the correlation of ambient vi-brations as this method is independent of earthquakesources. Using this technique co-seismic velocity changeswere observed in fault zones for example in Japan (We-gler and Sens-Schonfelder , 2007; Wegler et al., 2009;Takagi et al., 2012; Hobiger et al., 2012, 2013, 2016),in California (Brenguier et al., 2008b; Wu et al., 2016),and China (Chen et al., 2010; Froment et al., 2013;Ober-mann et al., 2014). Applying the deconvolution betweenborehole and surface records of the mainshock waves ofthe Tohoku-Oki earthquake, Nakata and Snieder (2011)observed the evolution of the velocity during the strongground motion. Since most of the coseismic changes areobserved within the fault areas that experience strongvibrations, and potentially also changes of the staticstress, there has been some debate about the origin ofthese velocity changes. The absence of any observationsof co-seismic velocity increases is an argument in favorof the dynamic stress during shaking as dominant causeof the observed velocity changes. Static stress changeswould lead to velocity perturbations with both polar-ities depending on whether the static stress change iscompression or dilation.

A further argument for dynamic stress as maincause of co-seismic changes comes from observationsof velocity changes at large distances from the faultarea, where static stress changes are far smaller thanthe dynamic stress during the passage of the seismicwaves. Lesage et al. (2014) analyzed velocity changesduring 15 years at Volcan de Colima and found – be-sides some relation with volcanic activity – a clear re-

A physics-based model of nonlinear elasticity 3

sponse of the velocity to several tectonic earthquakes atconsiderable distance from the volcano. The co-seismicvelocity changes are confined to the shallow layers of thevolcanic edifice. Richter et al. (2014) and Gassenmeieret al. (2016) analyzed noise data recorded by a seismicstation in northern Chile that appeared to be sensitiveto both static stress changes induced by daily and an-nual temperature variations as well as to dynamic stresscaused by seismic waves. After removing the seasonalsignal and a linear trend, Gassenmeier et al. (2016) suc-ceeded in predicting the velocity changes from daily in-tegrated absolute acceleration at the station over abouteight years that include two major earthquakes and alarge number of transients resulting from smaller lo-cal and regional earthquakes. With this model it wasdemonstrated that the velocity response does not de-pend on the earthquake source but is characterized bythe local shaking at the observation site. Brenguier et al.(2014) mapped the velocity decrease after the Tohoki-Oki earthquake in Japan and found an irregular dis-tribution that neither correlates with the intensity ofshaking nor the static stress change. However, by scal-ing the observed velocity changes by the intensity of theshaking, Brenguier et al. (2014) estimated the suscep-tibility of the velocity to shaking. This clearly showeda correlation with the active parts of the volcanic frontin northern Japan, and Brenguier et al. (2014) arguethat the overpressure of fluids in the volcanoes leadsto the pronounced susceptibility. A velocity reductionlocated below 5 km depth in the southern part of theNicoya Peninsula was also attributed to the presence ofoverpressured fluids in the subduction zone (Chaves andSchwartz , 2016).

Co-seismic velocity changes are followed by a recov-ery process during which the velocity increases again to-wards its original value. This recovery has been observedin nearly all studies that report a co-seismic decrease.The recovery is nonlinear in time and has been modeledwith exponential functions (Hobiger et al., 2012, 2013;Gassenmeier et al., 2016) that either increase asymptot-ically to the pre-seismic level or maintain a permanento↵set. Gassenmeier et al. (2016) used exponential func-tions with recovery times depending on the excitationand noted that the fast recovery immediately after theperturbation is not well modeled together with the longtail of the recovery process that can take several monthsto years if a single recovery time is used.

In summary, there are on the one hand observationsof positive and negative seismic velocity changes relatedto static stress changes resulting from the pressurizationof volcanic reservoirs or changes of surface temperature.On the other hand, there are numerous observations ofdecreasing velocity during the shaking by seismic wavesand subsequent recovery. So far, attempts to model thisbehavior have been purely phenomenologically.

2.2 The Benefit of Understanding the Physicsof Seismic Velocity Changes

Despite the number of seismological observations thereis no comprehensive understanding of the physics in-volved in the changing velocities. It is clear that staticchanges of the ambient stress increase or decrease theporosity of a rock and thereby change the velocity ofseismic waves. But why does a fast excursion of thestress during a seismic wave cycle leave the medium in adi↵erent state even though the stress is the same after-wards? Why does a dynamic excitation always lead toa velocity decrease ? What happens during the recoveryprocess? Those questions are interesting on their own,but they also have practical implications.

The seismic velocity changes reflect internalchanges in the material. Likely these changes a↵ectother properties beyond the seismic velocity of the ma-terial. For static stress changes it is known that varia-tions of the velocity go along with changes of the electricconductivity (Kaselow and Shapiro, 2004) because theclosure of compliant porosity does not only increase themodulus but also reduces the resistivity. Elkhoury et al.(2006) observed an increase in hydraulic conductivityduring earthquake shaking by analyzing the phase ofthe tidal response of the water level in wells. They ob-served an up to threefold increase in conductivity scalingwith the peak ground velocity at the well location ratherthan event magnitude or epicentral distance. Marc et al.(2015) observed a systematic increase of the landslideprobability after earthquakes in Japan and Taiwan. De-spite the co-seismic mass movements, the landslide ac-tivity remained elevated for several months to years af-ter the earthquake, indicating that the earthquake shak-ing damaged the material and made it more susceptibleto triggers like rainfall. These examples indicate thatchanges of the seismic velocity are only one specific ex-pression of a more fundamental change within the ma-terial. However, seismic velocity is an expression of thismaterial change that is particularly easy to monitor incontrast to conductivity and material strength with therelated landslide activity. A physical understanding ofseismic velocity changes might therefore help to developa more fundamental understanding of co-seismic pro-cesses in the material with consequences for di↵erentphysical properties.

Existing phenomenological models to explain seis-mic velocity variations in the field are restricted to fit-ting exponential functions to the observations (Hobigeret al., 2012, 2013; Gassenmeier et al., 2016). They areambiguous with respect of causative mechanisms. Lab-oratory experiments o↵er more control and allow formuch more detailed observations. We therefore includethe information from laboratory experiments in the de-velopment of the model for damage and recovery. Themain assumption of this article is that we can trans-fer knowledge from processes investigated in laboratorysamples of centimeter scale over periods of a few sec-


onds to a few hours to the field scale of kilometers andtime periods of month or years. We return to this insection 5.

2.3 Laboratory Experiments

In the acoustics community the observational capabil-ities are much wider than in field seismology and de-tailed models have been developed to explain temporallyvariable velocities. Typical experiments are quasi-staticloading tests (Guyer et al., 1995, 1997), resonant barexperiments (Guyer et al., 1998; TenCate et al., 2000,2004; Pasqualini et al., 2007) or cross modulation ex-periments (Zaitsev et al., 2005, 2014). In quasi-staticload experiments one directly observes the stress strainrelation over slow cycles of increasing and decreasingstrains on a sample in a load frame. These experimentshave revealed the nonlinear hysteretic behavior of rocksthat show di↵erent moduli depending on the strain his-tory. On the other hand resonant bar experiments areused to observe the dependence of the elastic moduli onthe amplitude of a dynamic excitation, which is usuallya harmonic vibration close to the resonance frequencyof the sample. Like in the quasi-static experiments theresonance experiments show a hysteresis with di↵erentbehavior depending on the amplitude history (Johnsonet al., 1996; TenCate et al., 2000). Additionally, the res-onant bar experiments allowed the investigation of thetemporal evolution of the modulus with a small ampli-tude signal after stopping the high amplitude vibration.The recovery process can be observed in these exper-iments over an extended time scale compared to seis-mological field observations. This led to the discoveryof so called slow dynamics (TenCate, 2011) that refersto a linear increase of the elastic modulus with the log-arithm of time. Similar observations have been madeusing cross modulation experiments that record the am-plitude variations of a probe wave induced by a pumpwave of higher amplitude and di↵erent frequency. Theelastic nonlinearity or NME materials expressed by (1)coupling of wave propagation at di↵erent frequenciesthrough cross modulation, (2) excitation of harmonics,(3) decrease of the modulus and (4) increase of attenu-ation, is sometimes termed anomalous fast dynamics. Ithas been shown to occur together with the slow dynam-ics (Johnson and Sutin, 2005). Detailed observations ofthe elastic nonlinearity are possible with a recently de-veloped experimental setup called dynamic acoustoelas-tic testing that are detailed in the next section.

2.3.1 DAET Experiments

Many author suggested that NME involves a multi-tude of processes that di↵er in their activation ener-gies (Prandtl , 1928b; Lyakhovsky et al., 1997; Vakhnenkoet al., 2006; Zaitsev et al., 2014). Especially the log(t)dependence of the recovery process is indicative of the

participation of processes with di↵erent time scales(Snieder et al., 2017). Resonant bar experiments andquasi-static load experiments provide insight into thecollective response of all contributing processes. Dy-namic acoustoelastic testing (DAET) is a method sim-ilar to a resonant bar experiment with the additionalpossibility to track the modulus of the material duringeach cycle of the strain (Renaud et al., 2012). While thetraditional setup measures the resonance frequency andthereby averages over the changing properties duringseveral strain cycles, DAET can resolve the dynamicsof the elastic properties during the change from com-pression over maximum strain rate to dilation and back.This makes it possible to separate processes dependingon their time scale. Fast processes will instantaneouslyadopt to changing conditions while slow processes donot respond to the changes within a cycle. The observeddynamics therefore reflects predominantly the processeswith timescales equal to the inverse of the excitationfrequency. This provides valuable information about thephysics of the individual processes that is otherwise hid-den in their collective response observed with resonantbar or quasi-static load experiments.

A series of DAET experiments show the detailedbehavior of di↵erent rocks (Renaud et al., 2012, 2013;Riviere et al., 2013, 2015), the shallow subsurface mate-rial (Renaud et al., 2014) and the pressure and frequencydependence (Riviere et al., 2016).

For the discussion in the end it is essential to un-derstand the principle of the DAET experiments thatseparate the perturbation process induced by a low fre-quency pump wave (up to several kHz) from the obser-vation typically performed with a low amplitude ultra-sonic pulse of MHz frequency. While the low frequencypump oscillation is longitudinal along the rock bar, theprobe wave is transmitted across the bar perpendicu-lar to the pump oscillation to sense a well-defined stateof the spatially varying strain. The main observationof DAET is the velocity of the probe wave as functionof time, or alternatively as function of the pump strainduring steady state oscillation. For easy comparison ofour model with the experimental observations we re-produce the relevant figures from Renaud et al. (2013)and Riviere et al. (2016) in Figures 1, 2 and 3. Figure 1shows the modulus variation calculated from the veloc-ity of the probe wave as function of pump strain forseveral experiments with di↵erent pump amplitudes. Inthis article we use positive strain for tension and neg-ative strain for compression as indicated in Figure 1.These nonlinear signatures represent the course of thematerial’s elastic modulus over a whole strain cycle. Toquantify the rather complex shapes of the nonlinear sig-natures, Renaud et al. (2013) fitted the relative changeof modulus �M/M0 by second order polynomials instrain "

�M/M0 = C

E

+ �

E

"+ �

E

"

2. (1)


Here M0 is the modulus of the relaxed sample. CE

is theo↵set of the modulus at zero strain during action of thepump. �

E

and �

E

are related to the classical quadraticand cubic nonlinearities, respectively. Here they repre-sent the average slope and curvature of the signaturesin Figure 1 as indicated by the black parabolas. Thesenumbers are illustrated for di↵erent amplitudes of thepump strain in Figure 2. Finally, Figure 3 (modifiedfrom Riviere et al., 2016) shows the amplitude of the ve-locity signal components at the fundamental frequencyand its even overtones for di↵erent excitation frequen-cies.

2.4 Existing Models

A review of early models of the nonlinear acoustic be-havior of rocks is included in Ostrovsky and Johnson(2001). The classical approach to the description of elas-tic nonlinearity is the extension of the stress strain rela-tion to higher orders in the strain (Landau et al., 1986).This accounts for atomic nonlinearity, but it fails to ac-count for the time dependence and hysteresis observedin NME. A phenomenological model for the hystere-sis and end point memory in quasi-static experiments isthe Preisach-Mayergoyz (PM) space (Guyer et al., 1995,1997) that uses a set of bistable hysteretic elements thatflip at di↵erent stress levels back and forth between twostates with di↵erent contribution to the material’s mod-ulus. Due to the lack of a relaxation mechanism thePM-space models fail to reproduce the slow dynamics.This has been fixed by including Arrhenius type tran-sition probabilities between the states in the PM space(Gusev and Tournat , 2005).

Lyakhovsky et al. (1997) presented a damage rhe-ology model for load experiments that did not focus onacoustic observations at small strains but leads to in-teresting results. The model uses a single damage stateparameter that a↵ects the elastic moduli of the dam-aged material. The kinetics of the damage parameter isbased on thermodynamic principles and leads to regimesof healing and fracturing depending on the strain invari-ant ratio.

A scalar damage parameter that scales the tension-compression asymmetry of the stress-strain relation ina cracked medium was used by Lyakhovsky et al. (2009)to describe quasi-static observations and the decreaseof the resonance frequency with increasing drive ampli-tude.

Vakhnenko et al. (2006) presented the soft ratchetmodel for longitudinal vibrations in a rock bar. Themodel uses the interaction between a fast and a slowsubsystem, where the fast system is responsible for thewave propagation and the slow subsystem represents theresponse of the cohesive bond system to the deforma-tion. The bonds break and are reestablished at di↵erentrates that depend on the applied stress. The soft ratchetmodel involves a number of time scales for breaking and

reforming the bonds. Vakhnenko et al. (2006) arrive at acomprehensive description of NME including the shapeof resonance curves and the amplitude dependent fre-quency shift, i.e. softening. The findings of the laterdeveloped DAET are not modeled by the soft ratchetmodel.

Lieou et al. (2017) modeled the material softeningobserved in granular material as the number of “sheartransformation zones” (STZ), which represent soft spotsin the medium. The number of these soft spots, whichreflect locations of strain concentrations depends on thedynamic strain amplitude and correlates with the mod-ulus reduction. This model can account for the modulusreduction with increasing dynamic excitation but doesnot explain the time dependency of the healing.

Attempts to model the nonlinear signatures ofDAET experiments were published by Gliozzi andScalerandi (2014); Pecorari (2015); Trarieux et al.(2014). Gliozzi and Scalerandi (2014) perform a wave-field simulation in a 2D medium that includs hystereticelements similar to the PM space. While this model re-produces the dynamic softening, it does not reproducethe shape of the loops including the orientation withincreasing modulus at maximum absolute strain. Themodel of Pecorari (2015), which is based on the interac-tion of dislocations with point defects, and microcracks,reproduces the softening and the shape and orientationof the DAET signatures reasonably well but exhibits adiscontinuity of the modulus at zero strain and does notreproduce the healing.

Trarieux et al. (2014) used a third order expansionof the complex elastic modulus to explain the nonlin-ear signatures of water, silicon oil and water saturatedglass beads. They are able to reproduce the observedhysteresis, and tension-compression asymmetry also ofhollow air-filled glass beads that behave strongly non-linearly similar to rocks. Even though this approach isuseful to quantify the nonlinear behavior it does notprovide insight into the physics and it fails to reproducethe dynamic softening as a significant component of themesoscopic nonlinearity expressed by the overall dropof the velocity in response to dynamic excitation. Anoverview of models that attempt to describe the elastic-ity and its relation to damage is given by Broda et al.(2014).

3 MODEL FOR DAET MEASUREMENTS

A large variety of lab experiments is available to studydi↵erent facets of dynamic material changes. None of theexisting theoretical models is able to describe all the rel-evant features of quasi-static load and resonant bar ex-periments including slow dynamic healing together withthe details of the DAET signatures. In this section weintroduce a model for the damage and relaxation that isaimed at a unified description of the observed phenom-ena in a model that can be interpreted physically. To


Figure 1. Variation of the elastic modulus of of Berea sandstone during a cycle of the pump wave (from Renaud et al., 2013).The di↵erent closed curves correspond to di↵erent experiments with di↵erent pump amplitudes.

this end we start from the details of the DAET signa-tures to derive constraints on the characteristics of theinvolved physical processes. This leads to a concept thatdoes not only allow to reproduce the DAET signatures,it also models the damage and healing processes.

Examination of the patterns in the DAET signa-tures in Figure 1 leads us to identify three independentcomponents of the nonlinear response. We first note thatthe DAET signatures in Figure 1 can be separated intothe di↵erence between the branches of increasing anddecreasing strain on the one hand and their average onthe other hand.

The average signal can be modeled with simplefunctions of the strain amplitude. It is composed of a lin-ear trend (green dashed line marked �M

l

in Figure 4) ofdecreasing modulus for increasing strain (tension) andan approximately Gaussian contribution of fixed widthwith a parabolic maximum in the center that flattensout towards large strain (red dash-dotted line marked�M

c

in Figure 4). Low strain experiments sense its cen-tral parabolic part that causes a high modulus at small-est absolute strain. Large amplitude experiments sensethe whole shape of the Gaussian which causes a bumpcentered at the unstrained situation.

The di↵erence signal (the loops in Figure 1) is morecomplicated. It is neither a simple function of strain norof strain rate. Empirically it approximately resembles a

bow tie shape illustrated in Figure 4 with the �M

s

-curve in black. There is a strong increase of modulusat maximum absolute value of strain and a decrease ofmodulus across the central part of vanishing strain. Im-portant is the orientation of the bow tie loop indicatedby the blue dots that mark the branch of increasingstrain. The resulting modulus of the material that isperturbed by the di↵erent e↵ects reads

M = M0 +�M

l

+�M

c

+�M

s

(2)

and is illustrated in Figure 4.Additional to these components that model the

shape of the nonlinear signatures there is a shift of thewhole signature towards smaller modulus with increas-ing amplitude of the strain signal. This overall modulusdecrease and the recovery result as a side e↵ect from theprocess that we suggest to model the bow tie shaped sig-nal component. In the following paragraphs we explainhow the di↵erent signal components are described math-ematically. Their physical interpretations is discussed insection 5.1.

We linearize the decrease of modulus with the strainin the DAET signatures and thus account for it by alinear function

�M

l

(t) = �A"(t) (3)

This linear dependence describes the softening of the


Figure 2. Amplitude dependence of di↵erent signal compo-

nents (from Renaud et al., 2013).

Figure 3. Frequency dependence of di↵erent harmonic signal

components (from Riviere et al., 2016).

rock as it changes from compression to extension. Inaddition, we add the following dependence of the mod-ulus on the strain

�M

c

(t) = B

✓exp

�"(t)2

2w2

�� 1

◆. (4)

This component of the modulus shapes its behavior inthe central part of the deformation curve for strain withabsolute value less than the width w. We give a physical

Figure 4. Schematic illustration of the components ofDAET signatures. �M is the combined e↵ect with an ad-

ditional o↵set of the mean modulus.

explanation for the behavior of �M

c

(t) in section 5.1.These contributions to the modulus are instantaneousand do not introduce additional dynamics.

The bow tie shape can be modeled together withthe total decrease of the modulus with a strain ratedependent weakening of the material. If there are twocompeting process of which one increases the modulus,and the other decreases the modulus proportional to theabsolute value of the strain rate, the balance of theseprocesses leads to the observed behavior. We present insection 5.3 a physical model that leads to such a be-havior which is empirically modeled here in a slightlysimplified way.

We propose that the relevant processes occur in theweak bond system at numerous independent microstruc-tures that we treat here in an abstract way independentof the physical details. We only suppose that these mi-cro structures can be in two di↵erent states with di↵er-ent contribution to the modulus of the material. This issimilar to the bi-stable contacts of Zaitsev et al. (2014),the shear transformation zones of Lieou et al. (2017) orthe adhesive bonds in the slow subsystem of Vakhnenkoet al. (2006).

Suppose the strain rate dependent perturbation ofthe modulus of the material is given as

�M

s

(t) = �CN(t) (5)

whereN is the fraction of damaged micro structures andC is a constant related to the number density of these


structures. Temporal changes occur because the stateof the micro structures can change over time due todamage and healing processes. We describe the tempo-ral evolution of the fraction of damaged microstructures(N

i

) as

dN

i

dt

=g|"|⌧

i

(1�N

i

)� 1⌧

i

N

i

. (6)

Subscript i indicates that there is a multitude of mi-cro structures that heal at di↵erent rates. N

i

varies be-tween 0 indicating the perfectly relaxed state when allstructures are healed and 1 when all structures of typei are broken. In equation 6, �⌧

�1i

N

i

accounts for thehealing over a time ⌧

i

. The term g|"|⌧�1i

(1 � N

i

) de-scribes excitation (damaging) of micro structures. Thisrate is proportional to the number (1 � N

i

) of undam-aged structures, as well as the strain rate |"| meaningthat no structures are damages when the medium is arest (" = 0). In this case the second term leads to ex-ponential recovery with time ⌧

i

. This strain rate andthe constant g normalize the characteristic time ⌧

i

forthe creation of defects. The average number density ofdamaged microstructures is given by

N =

Pi

1⌧iN

i

Pi

1⌧i

, (7)

where the pre-factor 1/⌧i

increases the weight of fastprocesses required by the log(t)-recovery (Snieder et al.,2017).

For a harmonic strain excitation the shape of theDAET signatures due to �M

s

results from a decreaseof N at maximum positive and negative strain whenthe strain rate vanishes. Because of equation 5 this de-crease of N(t) increases the modulus. For smaller abso-lute strain the motion damages the material faster thanit heals leading to a decrease in modulus during tran-sition from positive to negative strain and vice versa.The superposition of the di↵erent components of coursecomplicates the the shape. The overall shape of the sig-nature depends on the amplitude of the strain. Whereasthe bow tie shaped loop scales in amplitude and strainrange with the pump strain, the width and amplitudeof the Gaussian are fixed. This causes a small peak inthe center of the loop for large strains while the bow tieshape is warped around the central part of the Gaussianfor small strain amplitudes. In the next section we illus-trate the predictions of this model and compare themwith experimental observations.

4 NUMERICAL EXPERIMENTS

4.1 Modeling DAET Experiments

Using equations 2 through 7 we model the response ofthe material to harmonic excitations of di↵erent am-plitude. The parameters used for this experiment aregiven in column exp. 1 of Table 1. Figure 5 shows the

Figure 5. Modelled perturbation of the elastic modulus dueto a 4.5 kHz sinusoidal strain excitation of 10�6 from 0.01 s

to 0.31 s. The appearance of the curve as a thick line results

from the high frequency fluctuations during the action of thepump shown in the closeup around 0.31 s when the pump is

turned o↵.

time series of the resulting modulus for an experimentanalogous to Renaud et al. (2013) where the sample isperturbed with a sinusoidal pump wave with a frequencyof 4.5 kHz and a peak strain of 10�6.

The modulus shows the same behavior as in the ex-periments of Renaud et al. (2013). During the action ofthe pump wave the modulus fluctuates around a shortterm mean that decreases rapidly in the beginning andslowly converges towards a lower value. When the ex-citation stops at 0.31 s the modulus re-increases - veryfast in the beginning and slower at later times.

Interesting detail about the processes that causechanges of the modulus are revealed by the short termresponse of the modulus captured in the nonlinear signa-tures during DAET. Following the procedure of Renaudet al. (2013) we model the signatures shown in Figure 6.The bottom curve in the rightmost panel corresponds tothe time series shown in Figure 5 around 0.3 s. Thesemodeling results are to be compared with the exper-imental observations reproduced in Figure 1. Qualita-tively the signatures reproduce the main features of theexperimental observations which are: (1) a reduction ofmean modulus with increasing maximum strain, (2) anorientation of the loops with increasing modulus at max-imum absolute strain, (3) the absence of cusps, and (4)a transition from linear signatures at large strains toparabola shaped signatures at small strains.

We quantify the information in the signatures fol-lowing Renaud et al. (2013) by approximating themwith second oder polynomials according to equation 1.The approximations are indicated in green in Figure 6.In comparison to the observations shown in Figure 2,Figure 7 shows the parameters of these polynomials asfunction of maximum pump strain. Note that the fig-ure displays the logarithm of the negative values. Theaverage slope of the modeled signatures is constant at�160 as determined by the parameter A (table 1) whileit slightly decreases from �80 to about �160 for strain


symbol meaning exp. 1 exp. 2 exp. 3

A slope of linear component 160M0 ˜ 0

B amplitude of central component 0.002M0 0 0w width of central peak 1.5⇥ 10�6 ˜ ˜

C amplitude of strain rate dependent change M0 ˜ ˜g damage constant 0.08s 8 s 1000s

⌧min

shortest time scale of healing 1⇥ 10�5s ˜ 0.01s

⌧max

longest time scale of healing 100s ˜ ˜

Table 1. Model parameters used for the simulation of the nonlinear behavior. ‘˜’ indicates that the value is identical to theprevious experiment listed to the left.

Figure 6. Modeled nonlinear signatures for a range of strain amplitudes between 2 ⇥ 10�7 (top in right panel) and 10�5

(bottom in left panel). The segments of increasing strain are plotted blue. Dashed boxes indicate the regions enlarged in theplot to the right. Green curves indicate second order polynomial fits of the signatures to quantify the shapes (figure 7).

above 10�6 in the experimental data. The curvature ofthe signatures is approximately constant at �2 ⇥ 108

below 10�6 strain in model and experiment and thenincreases by about two orders of magnitude for largerstrain. For the smallest strain the estimated curvature ofthe modeled signature is slightly positive and does notplot in these axis. The decrease of the mean modulusexpressed by the parameter C

E

changes approximatelylinear in the logarithmic axis from 2⇥ 10�5 to 10�2 inthe experiment while it covers the range from 10�4 to10�2 in the model.

Since we did not attempt to estimate best fittingparameters in a formal inversion process we concludethat the model reproduces the observations reasonablewell.

4.2 Frequency Dependence

The time dependence of the damage and healing pro-cesses leads to a dependence of the response on exci-tation frequency. To compare our model predictions to

laboratory measurements we conduct numerical experi-ments analogous to Riviere et al. (2016) in the frequencyrange from 0.2 to 200 Hz. In this experiment 2 we usethe parameters given in column exp. 2 of Table 1 withthe di↵erence to experiment 1 being the neglect of �M

c

(B = 0) that is not observed by Riviere et al. (2016)and a larger damage constant g. To obtain results thatcan be compared to the observations of Riviere et al.(2016) we decompose the waveforms during the station-ary phase of the oscillation into their harmonic compo-nents. We thereby obtain the amplitude of the velocitysignal at the excitation frequency f1 and its overtonesf2 . . . f7. Additionally there is the overall decrease ofthe average modulus indicated as f0. This procedure isrepeated for various excitation frequencies and we plotthe amplitudes of the harmonic components as functionof frequency in Figure 8.

Compared with the laboratory observations repro-duced in Figure 3 the numerical experiment reproducesthe first order behavior. Except for the fundamental fre-quency component f1 that does not show any frequency


Figure 7. Amplitude dependence of the nonlinear signa-tures. C

E

, �E

and �E

are the coe�cients of polynomial ap-

proximations of the signatures.

Figure 8. Frequency dependence of the nonlinear responseas amplitude of harmonic signal components.

dependence, the amplitudes of all other components in-crease linearly with logarithm of frequency. The ampli-tude of the decrease of the mean velocity (f0) is largestfollowed by the even overtones with decreasing ampli-tudes for increasing oder of the overtone. Amplitudes ofthe odd overtones (f3, f5 and f7) are at least an order ofmagnitude below the even harmonics and are not shownin figure 3. Absolute amplitudes are not well matchedbut could be adjusted with the model parameters A andC.

4.3 Slow Dynamics

To illustrate how our model reproduces the typical ob-servations of slow dynamic recovery we perform a thirdnumerical experiment. Since this experiment focusseson the evolution of the mean modulus we use an exci-tation that avoids fast oscillations as seen in Figure 5during the action of the pump. We achieve this by us-ing a pump waveform that keeps the damage term inequation 6 constant, i.e. by using a triangular waveformwith |"| = const as indicated in the inset of Figure 9A.We also set parameters A and B to zero as �M

l

and�M

c

do only cause fast oscillations and do not con-tribute to the long term dynamics. Other parametersof experiment 3 given in Table 1 are adopted to repro-duce the long term behavior in analogy to (TenCateet al., 2000). The excitation lasts for 80 s and the re-covery is followed for 500 s. To illustrate the e↵ect ofthe minimum time scale we use a minimum relaxationtime ⌧

min

= 0.01 s. The saturation of the damage phasecan be observed because we use a much higher damageconstant of g = 1000 here. Figure 9A shows the time de-pendence of the modulus as a function of the linear timescale. Red part shows the damaging phase during the ac-tion of the pump. The recovery process is shown with ablack line. Figures 9B and C show the damage and re-covery phases on logarithmic time axis respectively witht0 and t1 being the start and end times of the excitationrespectively. The minimum and maximum time scalesof the healing process are indicated in C. Change of themodulus during conditioning (approximately) and re-laxation is linear on the logarithmic time scales similarto observations by TenCate et al. (2000) and Johnsonand Sutin (2005). As in the examples of Snieder et al.(2017), the logarithmic time behavior occurs for times⌧

min

< t� t1 < ⌧

max

.

5 DISCUSSION OF THE MODEL RESULTS

In the previous section we have shown the predictionsof our model in conditions comparable with publishedexperiments. In particular, we looked at the long termtemporal dynamics (slow dynamics), the decrease of themean modulus after dynamic excitation of di↵erent am-plitudes, the nonlinear signatures of harmonic excitation(DAET) and the frequency dependence of the nonlinear-ity.

In the following we discuss the predictions of themodel in comparison with results of laboratory exper-iments. We restrict the discussion to comparison withmeasurements in Berea sandstone but the set of mea-surements on di↵erent rocks presented by Riviere et al.(2015) allows to judge the ability of our model to ex-plain the behavior of di↵erent materials. Compared tothe main features we aim to reproduce, the di↵erencesbetween di↵erent rock are minor.


Figure 9. Slow dynamics of the damaging and relaxation phases. Modulus change during the damage phase is indicated in redwhile relaxation is plotted black. A) Modulus change during experiment 3 on a linear time axis. Inset shows the triangular strain

curve of the pump used to avoid oscillations. B) and C) show the damage and relaxations phases respectively on logarithmic

time axes with t0 being the start and t1 the end times of the excitation, respectively. ⌧min

and ⌧max

in C) indicate the rangeof relaxation times in this numerical experiment.

5.1 Nonlinear Signatures

The main properties of the nonlinear signatures thatour model has to explain are the decrease of the meanmodulus with increasing amplitude of the strain oscilla-tions, the hysteresis, the absence of cusps at maximumstrain and the orientation of the hysteresis loop withincreasing modulus at maximum absolute strain. Let usfirst consider the shape of the signatures as the mostprominent feature. The good match between the ex-perimental observations of Renaud et al. (2013) in Fig-ure 1 (also Riviere et al., 2013, 2015) and our modelresults from the superposition of three components inour model (Figure 4), two of them are strain dependentand one depends on strain rate and has a time depen-dence.

The linear contribution can be linked to the pres-sure dependence of the wave velocity through the clos-ing of compliant or soft porosity of elongated cracks andcontact vicinities (Athy , 1930). Shapiro (2003) showedthat the compliant porosity �

c

is given as

�

c

= �

c0 exp(�✓�

s

P ) (8)

where P = P

c

� P

p

is the e↵ective pressure resultingfrom confining pressure P

c

and pore pressure P

p

. �c0

is the compliant porosity at zero e↵ective stress and�

s

is the sti↵ compressibility of the rock in the hypo-thetical condition of all closed compliant porosity. Thepiezosensitivity ✓ describes the sensitivity of the ma-terial to stress changes. For constant pore pressure thisconsideration leads to an exponential dependence of thewave velocity on the confining pressure (Shapiro, 2003),which can be linearized for the small fluctuations of thedynamic excitation in DAET giving rise to the linearcomponent �M

l

.The bell-shaped component �M

c

is more di�cultto interpret. We suggest that it results from shear mo-tion on the rough interface of a crack. The shear motionoccurs in a rock sample due to internal heterogeneity

even if the macroscopic excitation is longitudinal. In thissituation, it is reasonable to assume that the roughnessis random but to some extent correlated on both sides ofthe crack for zero strain. In the unstrained situation thesurfaces fit into each other like to puzzle pieces, maxi-mizing the contact area and consequently the modulus.If the crack is sheared the fit is less good and the contactarea decreases due to the progressive mismatch until itis finally restricted to the tips of the surfaces leading toa decreased modulus. For larger shear, beyond the cor-relation length of the surface roughness, the two sides ofthe crack are uncorrelated and the average contact areadoes not depend on the strain any more. This behavioris well captured by the bell-shaped strain dependence.However, this component of the signatures appears to beless important for observations in other materials thanBerea sandstone in the kilohertz regime (Riviere et al.,2015, 2016).

In our model the signal component at the excita-tion frequency originates only from the linear compo-nent �M

l

, because �M

c

and �M

s

are linked to shearwhich is symmetric for positive and negative strain andthus excite only the even harmonics at two, four or sixtimes the excitation frequency. Since �M

l

has di↵erentsign for compression and dilation it must result fromcompression and it should thus disappear for shear ex-citation. This was indeed observed by Lott et al. (2017)as vanishing of the fundamental frequency componentfor shear excitation.

O↵set of mean modulus, hysteresis, absence ofcusps and the orientation of the loops, which are com-mon features in all the DAET observations, are all re-lated to the strain rate dependent damage and healingmechanisms of our model included in �M

s

. The dynam-ics results from structures with the permanent tendencyto heal, i.e. establish sti↵ connections, and the propen-sity to be damaged when strained. The interplay of thesetwo mechanisms leads to an increase of the modulus


whenever the material is at rest. When the material isbeing deformed the reduction of the modulus dependson the relative rates of damage and healing. Faster de-formation leads to more severe damage. In this contextit is natural that the dynamic signatures do not showcusps because at the maximum absolute strain the ma-terial is not damaged and the healing process leads to asmooth increase of modulus until the strain rate reversesand damage starts again. This defines the orientation ofthe loops and leads to the observed hysteresis.

The absence of cusps appears to be in contrastwith the quasi-static observations of Guyer et al. (1997)(Ostrovsky and Johnson, 2001; TenCate, 2011; Riviereet al., 2016). How does our model reconcile these ob-servations from quasi-static and dynamic experiments?A detailed modeling of the quasi-static experiments isbeyond the scope of this paper but we sketch the simplemechanism that explains both the fast DAET and thequasi-static experiments. Obviously, the speed and thestrain range at which the experiments are performeddi↵er by orders of magnitude. Typically, the stress inquasi-static experiments is linearly increased over a timeof several minutes. When the peak stress of one cycle isreached, the stress evolution is reversed immediately re-sulting in a triangular driving. So one could think thatthe fast reversal of the stress curve prevents the dam-age from healing and thus produces the cusp in the loadexperiments while the sinusoidal change of the pumpstress in the DAET experiments lets the material be atrest during the stress peaks which allows the healing tosmooth the cusps. However, this possibility is ruled outby Claytor et al. (2009) who show with very slow loadexperiments that the hysteresis vanishes while the cuspsremain.

In quasi-static load experiments hysteresis leads tothe open curves in the stress - strain behavior that fol-low di↵erent paths for increasing and decreasing strain.Since there is no possibility to measure the stress di-rectly in DAET experiments one uses the wave velocityof a probe wave to sense changes of the elastic modu-lus. Also the velocity follows open curves in the strain- modulus space. However, the instantaneous modulusM cannot be interpreted as the one that relates themacroscopic stress � and strain ". Given the orienta-tion of the loops, such an interpretation would lead tonegative dissipation when integrating the strain energydensity E = 1/2M"

2 over the DAET loop. Our modelo↵ers a simple mechanism to reconcile the orientation ofthe loops with the fact that energy is dissipated ratherthan released by the material when we turn to the phys-ical mechanisms that might cause the strain rate depen-dence in section 5.3.

5.2 Frequency and Pressure Dependence ofthe Nonlinearity

Our model qualitatively reproduces the experimentallyobserved frequency dependence by Riviere et al. (2016).The amplitude of the velocity variation at the frequencyof excitation (f1) is independent of frequency. Oscilla-tion at f2, f4 and f6 as well as the static o↵set (f0) varywith frequency according to a power law �c/c = �fµ.The harmonics at three, five, and seven times the exci-tation frequency have drastically lower amplitude com-pared to the even harmonics as observed in the exper-iment, but follow the same power law. The power lawexponent is µ ⇡ 0.5 in our simulations compared toµ ⇡ 0.16 found experimentally by Riviere et al. (2016).This di↵erence might indicates that the damage andhealing processes are not as simply related to the strainrate as assumed in our simple model. It is likely thatthe damage process does not solely depend on strainrate but to some degree also on strain. This could lowerthe frequency dependence.

We discussed the pressure dependence of the wavevelocity as the origin of the linear component in theDAET signature. However, the closure of compliantporosity does not only alter the wave velocity directly.It also removes the weak parts of the bond system thatare thought to be responsible for the nonclassical non-linearity which leads to hysteresis and slow dynamics(Johnson and Sutin, 2005). We thus propose that theparameters A, B and C that scale the nonlinear compo-nents in our model all depend on the compliant porosityas described by equation 8. This coincides exactly withthe functional form used by Riviere et al. (2016) to de-scribe dependence of nonlinearity on confining pressurewhere the characteristic pressure P0 of Riviere et al.(2016) equals 1/✓�

s

of Shapiro (2003).

5.3 Physical Interpretation of the Strain RateDependence

In the previous sections we introduced a mathematicalmodel that accounts for a number of observations fromDAET experiments. Most importantly the shape of theDAET signatures, the decrease of the mean modulusand the orientation of the hysteresis loops can be mod-eled using a strain-rate proportional damaging process.However, there is no explanation of what this physicalprocess could be. In fact, we do not intend to describethe microscopic mechanism precisely, because it is un-likely that there is a single microscopic mechanism re-sponsible for macroscopic observations. But we describein this section one possible mechanism with the dynam-ics required to explain the DAET results. We call it theself-healing crack.

There are di↵erent physical processes which couldestablish connections in the bond system that can bebrocken by deformation. Those are for example capillarybridges (Bocquet et al., 1998; Barel et al., 2012), chemi-


cal bonds or van der Waals forces (Li et al., 2011; Tianet al., 2017; Liu and Szlufarska, 2012), adhesive con-tacts (Barthel , 2008), or fibers of minerals that grow incracks as observed in-situ by Vanorio (2015), Hilloulinet al. (2016), Wiktor and Jonkers (2011) and Vanorioand Kanitpanyacharoen (2015). These processes can me-chanically connect the opposite sides of a crack, therebyincreasing the modulus – or the connection can be bro-ken leading to a reduced modulus. A demonstration ofsuch a process on the atomic level can be found in Liet al. (2011).

Consider the surfaces of the two opposing sides ofa narrow crack. This might be a tiny crack or portionsaround the contacts or at the edges of a larger crack.Such a contact is illustrated in Figure 10A. When thecontact is at rest connections are formed, which connectthe two sides of the contact as illustrated by the con-necting lines. Our hypothesis here is that even thoughit takes days until self-healing can be observed directlywith a microscope as new minerals, the process of creat-ing bonds starts immediately upon formation of a con-tact (Li et al., 2011). This process is faster for smallercracks and at the smallest scale the healing a↵ects thematerial at the frequency of the strain changes in theDAET experiments (several kHz).

If the crack is deformed, connections are strainedand break while new ones are formed at the current de-formation state (Figure 10B). At peak strain of an har-monic deformation the material is at rest for a momentmeaning that no connections are damaged but healinggoes on. This leads to hardening of the material aroundthe peak strain (Figure 10C). When the strain rate re-verses the deformation damages connections which leadsto fewer connections in place when the material passesthrough the zero-strain state (Figure 10D). The modu-lus that is measured by the high frequency probe wavein a DAET experiment is proportional to the number ofconnections across the crack – no matter which portionsof the surfaces they connect.

Figure 11 shows a numerical simulation of DAETof such a crack as an ensemble of a large number ofcrack elements without the strain dependent compo-nents �M

l

and �M

c

. Crack elements are the pillarson the bottom side of the crack in Figure 10. The prob-ability of such a crack element to form a connectionto the opposite side is constant in time and given bythe healing time ⌧

i

in equation 6. For a small simula-tion time step �t this probability is P

c

= �t/⌧

i

andthe connections are formed towards the closest point onthe opposite side at the current deformation state. Theprobability of a connection to break increases with thestrain of that connection, i.e. the deformation betweenthe condition under which the connection was formedand the current strain. For a small time step �t in thesimulation the probability of a connection to break isP

b

= �tF�". Where �" is the strain of the connec-tion and F is a constant. In Figure 11 we simulate the

A

B

C

D

Figure 10. Illustration of contact with fragile connec-tions formed by chemical bonds, mineralization or capillary

bridges. The contact is perfectly healed in (A) and deformed

via (B) to peak strain illustrated in (C). When returning tothe initial position the contact maintains damage (D).


Figure 11. Nonlinear signatures of the self healing crackmodel.

motion of the top part of the crack shown in Figure 10over the bottom and track the connections of every pil-lar: at which strain is a connection established, how is itstrained (�"), and when does it break. The result of thesimulation is the total number of connections present asproxy for changes of the material’s modulus observedby the DAET probe wave. Parameters are chosen suchthat P

b

and P

c

are of similar magnitude for �" = "

max

.Qualitatively the model of the self-healing crack is

able to reproduce the main features of the DAET ob-servations such as the decrease of mean modulus, andthe bow tie shaped loops with their orientation.

The self-healing crack model gives insight why inthe damage in the mathematical model in section 3 isrelated to strain rate while intrinsically damage occursdue to strain. Due to the realignment of connectionsduring the healing phase the bond system adopts to anystrain and new damage is only introduced by a changeof strain, i.e. strain rate.

The model also provides additional insight intoDAET measurements. On the one hand any connectioncontributes to the modulus seen by the probe wave inDAET. On the other hand the contribution of a con-nection to the macroscopic stress involved in the pumpwave depends on its restoring force which in turn de-pends on the strain state under which it was created.The connections in Figures 10C and D that bridge thecrack perpendicularly do not contribute to the stressinvolved in the pump wave of a DAET experiment orthe static stress in a load experiment as they do notcontribute to the restoring force. But they contributeto the modulus seen by the probe wave because theycounteract any additional deformation (e.g. the probe).Therefore the modulus seen by the probe wave must notbe confused with the modulus involved in the macro-scopic deformation. The modulus observed with theprobe wave in the DAET signatures is thus not suit-able to draw conclusions on the energy dissipation ofthe pump oscillation. Since the connections are estab-lished at the current deformation state the healing pro-cess does not increase the strain energy. However, con-nections break when strained. Whenever a connection

breaks, strain energy is lost, likely as heat for smallstructures (Prandtl , 1928b) or as damage-related ra-diation for macroscopic structures (Ben-Zion and Am-puero, 2009). This mechanism provides a link betweendeformation, velocity reduction and attenuation of thelow frequency excitation.

Hysteresis is only related to processes with charac-teristic times of healing (⌧

i

) larger than the time scaleat which changes are induced in the experiments. Fasterprocesses adopt instantaneously (compared to the ex-perimental time scale) to any new strain and do notcontribute to hysteresis. Since the number of contribut-ing processes decreases for slower experiments the hys-teresis in slowly performed load experiments vanishes,as observed by Claytor et al. (2009).

The proposed mechanism, that healing occurs byestablishing connections across a contact at the currentstrain state, can also explain the absence of cusps inDAET and their presence in quasi-static load exper-iments. In load experiments the observed stress cor-responds to the applied strain as there is no distinc-tion between excitation and observation. So naturally,at maximum strain the modulus does not increase in thehealing process because the connections are establishedat exactly this strain and do not increase the restoringforce. (Actually the modulus can decrease as strainedconnections can continue to break leading to plastic de-formation.) Since the healing process does not alter themacroscopic modulus relating stress and strain in theload experiment there are cusps at the turning pointsof the load protocols. In contrast the probe wave in aDAET experiment will not see cusps as illustrated inFigure 11.

5.4 Slow Dynamics

Slow dynamics is a term usually used to refer to therelaxation and associated sti↵ening of a material aftersome perturbation. Typically this is observed as the in-crease of the mean modulus towards its original valueafter a damaging event. Slow dynamics has been ob-served using resonant bar experiments (TenCate et al.,2000; Johnson and Sutin, 2005; Riviere et al., 2013) asthe linear increase of the modulus with logarithmic time.In fact, the term slow dynamics is slightly misleading,as this e↵ect is indeed slow at late times, but it is fastin the beginning. Both regimes are di�cult to observeexperimentally and it is usually for intermediate timewhere the modulus changes proportionally to the loga-rithm of time. Johnson and Sutin (2005) observe thatalso the conditioning phase during the dynamic excita-tion follows a log(t) dynamics.

The log(t) dependence has been discussed recentlyfor example by (e.g. by Snieder et al., 2017) and it hasbeen suggested that it results from the simultaneousaction of variety of exponential processes with di↵er-ent timescales (TenCate et al., 2000; Vakhnenko et al.,


2006; Amir et al., 2012). Alternatively Brey and Prados(2001) showed that hierarchical models which containdi↵erent levels of similar processes that are activated insuccession also lead to similar behavior (Nabarro, 2001).

A spectrum of processes is explicitly incorporatedin our model by including a range of healing times ⌧

i

inequation 6. The observed linearity of the recovery curvein Figure 9 on the logarithmic time scale between theminimum and maximum healing times is thus built-inand does not reveal new insights into the physical mean-ing of the log(t) dynamics of the damage and healingprocesses.

Our work does not provide new experimental evi-dence for the physical cause of the log(t) behavior. Butthe close relation between damage and healing processesleads us to speculate about the reason for the scal-ing behavior. Under the term crackling noise, a num-ber of physical phenomena have been investigated thatrespond to slow external forces with discrete events(Sethna et al., 2001; Dahmen and Ben-Zion, 2009; Kazet al., 2012; Salje and Dahmen, 2014). Such phenomenaas Barkhausen noise of magnetization jumps, collapse ofporous materials, stock market fluctuations, fracture indisordered media or the sound emission of candy wrap-pers share the broad range of event sizes with whichthey respond to the forcing. Also, relaxation phenom-ena in the absence of external forcing show a variety ofinvolved scales that lead to the observation of log(t) dy-namics in a variety of systems (Amir et al., 2012). Thesesystems develop universal scaling relations between thesize of activated structures and their frequency of oc-currence.

It is not perfectly understood when and how thesystems produce jerks with broad scaling relations. Theearthquake process with fractures ranging from acousticemission in lab samples to catastrophic earthquakes iscertainly one of the most prominent examples with theGutenberg-Richter law (Gutenberg and Richter , 1949)and the Omori law (Utsu et al., 1995) as well-knownscaling relations for the spatial and temporal distribu-tion of events.

Since the damage in the self-healing crack modelproposed above occurs by shear deformation along weakinterfaces in the bond system (micro faults) it bearsclear similarity with the fracture process. In fact, thiscan be seen as the smallest size of the fracture scale justabove the discrete atomic level. We hypothesize that themacroscopically observed damage is created in discreteslip events along such micro faults with a frequency-sizescaling similar to the crackling noise of acoustic emis-sions and earthquakes. Actually these slip events couldpopulate the Gutenberg-Richter law for weak events.We speculate that the distribution of healing times atthe origin of the log(t) recovery is related to the samefrequency-size distribution of activated structures. Thiscould explain why the log(t) recovery appears as univer-sal as the Gutenberg-Richter law even though healing

involves a variety of processes with intrinsically di↵erenttime scales. In fact, the earthquake process itself withits relaxation during the aftershock phase also shows thelog(t) dependence in the postseismic deformation seenby GPS (Avouac, 2015) and laboratory friction experi-ments (Nagata et al., 2008).

An important property of our model is the rapidexcitation. Similar to the relaxation process, the dam-age or conditioning phase is approximately linear on alogarithmic axis, but it acts faster than the recovery asseen in Figure 9. This directly follows from equation 6because the characteristic time of the damage processis ⌧

i

/(1+ g|"|) compared to the time ⌧

i

of healing. Thisproperty of our model corresponds to the observationof a relaxation process that is linear on a log(t) axis fora period much longer than the duration of the excita-tion. If damage and healing rates were equal, the healingwould be finished after a period equal to the duration ofthe dynamic excitation. Measurements during the con-ditioning and relaxation phases performed by TenCate(2011,Figure 3), Johnson and Sutin (2005) and seismo-logical observations of velocity recovery for years aftera minute-long earthquake (Richter et al., 2014), clearlyindicate that damage is indeed faster than recovery.

The disparity of damage and healing rates seemsto be a particular property of the elastic nonlinearityin contrast to other relaxation phenomena (Amir et al.,2012) where the duration of the relaxation process iscorrelated to the duration of the excitation. This re-flects the fact that damage and healing, i.e. the destruc-tion and creation of connections in our model are di↵er-ent processes in contrast to e.g. changing magnetizationfrom one direction to the other.

6 CONCLUSIONS

Inspired by the wealth of information in the observa-tions of DAET experiments we have developed a simplemathematical model to explain variations of the seis-mic velocity in response to dynamic perturbations. Themodel consists of a material that contains structureswhich are broken by deformation and closed by somechemo-physical process with a range of healing rates,which continuously hardens the material. The impor-tant element is that the healing involves a range of timescales including the fast time scales of the kilohertz vi-brations used in DAET. Healing that occurs at non-zerostrain freezes this strain state which allows to modelcreep.

The decrease of the modulus for the same excita-tion will be smaller when the material is already dam-aged because there is a finite number of connectionsonly and broken connections cannot be damaged anyfurther. The strongest change will always be observedin the best healed conditions. The response of the ma-terial to a unit excitation does therefore depend on thestate of the material, i.e. the position of the material on


the continuous scale from perfectly healed to completelydamaged as quantified by the variable N in our model.

Our model bears a number of similarities with mod-els of friction (Popov et al., 2012). Most importantly itinvolves shear motion. The healing if the system is atrest increases friction as observed by Li et al. (2011) dueto the formation of bonds across a silica asperity. Thisis a process known as frictional aging and is discussedas a physical origin of rate and state friction. It pro-vides an explanation for the velocity dependent friction(Popov , 2010). As healing in our model, the frictionalaging exhibits log(t) dynamics (Li et al., 2011; Frye andMarone, 2002; Nagata et al., 2008). In fact our modelis similar to a model by Prandtl (1928b) for internalfriction as explanation for elastic hysteresis, aftere↵ectsand viscosity. The additional assumption in out modelis, that connections/bonds do not only hop to closer lo-cations when a contact is sheared, they can also remainbroken for some time before creating a bond to a newlocation.

For seismological observations of the seismic veloc-ity, i.e. the mean modulus, without detailed DAET in-formation, our description allows to model the evolutionof the seismic velocity for a series of excitations. Thisshould allow to infer values for the parameters C, g, ⌧

min

and ⌧

max

which constitute a new set of material param-eters with physical interpretations for dynamic loading,the nonlinearity between weak and strong motion andmaterial strength.

The governing parameter of the reversible damageprocess is the strain rate. This explains observations ofdecreasing velocities during slow shear motion (Clarkeet al., 2013; Mainsant et al., 2012; Rivet et al., 2011).The two mechanism of mechanically closing compliantporosity by static stress (�M

l

) and of dynamically cre-ating/destroying bonds/connections explain the posi-tive and negative velocity changes in response to staticstress changes at volcanoes (Sens-Schonfelder et al.,2014) and the exclusively negative changes due to dy-namic perturbations (Brenguier et al., 2014). Any tran-sient deformation that causes shear of contacts destroysconnection and softens the material even if the stain iszero afterwards. On the other hand a shear deformationcan never create connections. Creation of connectionsi.e. healing is thermally driven and it is fastest whenthe material is at rest without any counteracting dam-age.

However, both e↵ects depend on the compliantporosity and we can expect that materials with highpiezosensitivity do also exhibit higher sensitivity to dy-namic deformation as observed by Richter et al. (2014).But additionally we should expect a dependence of thedynamic e↵ect on the chemistry of the rock and theavailability of water as observed in friction experiments(Li et al., 2011; Frye and Marone, 2002).

We expect the damage and healing processes to oc-cur in the weak parts of the bond system, i.e. in the com-

pliant porosity which depends of the e↵ective stress. Thedamage and healing process is therefore strong near theEarth’s surface were the confining pressure is low. Dueto pore pressure counteracting the confining pressure insaturated materials the damage and healing processeswill also be relevant in reservoirs, volcanoes (Brenguieret al., 2014; Lesage et al., 2014) and subduction zones(Chaves and Schwartz , 2016). It might be an impor-tant process for dynamic triggering of earthquakes (Hillet al., 1993; Brodsky and van der Elst , 2014) or vol-canic eruptions (Watt et al., 2009; Nishimura, 2017) andthe damage and healing process might be an importantagent in the evolution of the aftershock activity.

ACKNOWLEGDMENTS

We gratefully acknowledge Frederik Tilmann, GeorgDresen for critical reviews of the manuscript and stim-ulating discussions with Yehuda Ben-Zion and MichelCampillo. No new data were used in producing thismanuscript.

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a physics-based model of nonlinear elasticity of rocks for

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