Relating strain localization and Kaiser effect to yield ...

Geophys J Int (2020) 221 2091ndash2103 doi 101093gjiggaa130

GJI Rock and Mineral Physics Rheology

Relating strain localization and Kaiser effect to yield surfaceevolution in brittle rocks

Hannah Gajst12 Eyal Shalev2 Ram Weinberger23 Shmuel Marco1 Wenlu Zh4 andVladimir Lyakhovsky 2

1The Department of Geophysics Tel-Aviv University Tel-Aviv 69978 Israel E-mail gajsthannahmailtauacil2Geological Survey of Israel 32 Yesharsquoayahu Leibowitz Jerusalem 9371234 Israel3Department of Geological and Environmental Science Ben Gurion University of the Negev Beer Sheva 84105 Israel4Department of Geology University of Maryland 237 Regents Drive College Park MD 20742 USA

Accepted 2020 March 8 Received 2020 March 8 in original form 2019 August 20

SUMMARYThe yield surfaces of rocks keep evolving beyond the initial yield stress owing to the damageaccumulation and porosity change during brittle deformation Using a poroelastic damagerheology model we demonstrate that the measure of coupling between the yield surface changeand accumulated damage is correlated with strain localization and the Kaiser effect Constantor minor yield surface change is associated with strong strain localization as seen in low-porosity crystalline rocks In contrast strong coupling between damage growth and the yieldsurface leads to distributed deformation as seen in high-porosity rocks Assuming that duringbrittle deformation damage occurs primarily in the form of microcracks we propose that themeasured acoustic emission (AE) in rock samples correlates with the damage accumulationThis allows quantifying the Kaiser effect under cyclic loading by matching between the onsetof AE and the onset of damage growth The ratio of the stress at the onset of AE to the peakstress of the previous loading cycle or Felicity Ratio (FR) is calculated for different modelparameters The results of the simulation show that FR gradually decreases in the case of weakcoupling between yield surface and damage growth For a strong damage-related couplingpromoting significant yield surface change the FR remains close to one and decreases onlytowards the failure The model predicts that a steep decrease in FR is associated with atransition between distributed and localized modes of failure By linking the evolving yieldsurface to strain localization patterns and the Kaiser effect the poroelastic damage rheologymodel provides a new quantitative tool to study failure modes of brittle rocks

Key words Fracture and flow Numerical approximations and analysis Numericalmodelling

1 INTRODUCTION

11 Pre-failure damage localization patterns

Brittle failure controls the stability of rock mass and produces rockdamage The process of brittle failure is different for low-porositycrystalline (eg granite φ lt 1) and high-porosity granular (egsandstone φ gt 15) rocks While the former tends to form highlylocalized discrete slip surfaces the latter may develop tabular zonesof deformation bands prior to the formation of slip surfaces (Fig 1)Both field and laboratory studies support this observation (eg Ay-din amp Johnson 1983 Lockner et al 1992 Antonellini et al 1994Weinberger et al 1995 Mair et al 2000 Fossen et al 2007 Gajstet al 2018)

Failure of crystalline rock is often described on multiple scalesas a localization process resulting in a discrete fault plane (egBen-Zion amp Sammis 2003) An example of localization on a plateboundary scale of kilometres was discussed by Marco (2007) whodemonstrated the gradual localization process of the Dead Sea FaultZone into a single shear zone over millions of years under a rela-tively constant regional stress field In contrast fault zones in poroussandstone are known to form packages of deformation bands eachaccumulating a small amount of displacement (mmndashcm) Thesezones can accumulate strains on the scale of up to several decime-tres prior to the formation of a slip surface (Aydin amp Johnson 1983Antonellini et al 1994 Fossen et al 2007) suggesting repeatedphases of localization and delocalization preceding faulting Ship-ton amp Cowie (2001) and Bernard et al (2002) demonstrated that the

Ccopy The Author(s) 2020 Published by Oxford University Press on behalf of The Royal Astronomical Society 2091

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Figure 1 A schematic drawing of the damage accumulation stages in crystalline rocks and porous sandstones Crystalline rocks show strong damagelocalization while porous sandstone may experience several localization cycles in the form of multiple deformation bands forming prior to the developmentof a slip surface (ie fault)

zone of shear bands could continue to grow even after the forma-tion of a slip surface thus exhibiting distributed strain accumulationafter faulting

In the laboratory acoustic emission (AE) is measured to probemicrocracking during brittle deformation Lockner et al (1992)recorded locations of AE activities during fault propagation in ax-ially loaded Westerly granite samples Their results show that afteran initial lsquoringing phasersquo where AE was recorded throughout thesample a fast localization process occurred This coincided withthe development of a localized fault zone and failure In the samestudy when subjected to similar experimental conditions Webersandstone samples underwent a slow transition from tabular dis-tributed zones of AE activity to a localized planer zone of AEactivity (Lockner et al 1992) The sandstone samples did not ex-hibit an initial ringing phase and their final deformation zones werethicker than those of the granite samples Results of experiments ofLocharbriggs sandstone (Mair et al 2000) demonstrated the gradualformation of a package of shear bands clearly relating the amount ofdeformation and the number of strands (ie bands) They suggestedthat each strand accumulates a finite amount of strain before a newstrand is formed This can be interpreted as multiple transitionsbetween modes of localized and distributed damage accumulationThe transition mechanism is difficult to explain since no evidenceof strain hardening was observed in this experiment

12 The Kaiser effect

The Kaiser effect (Kaiser 1953) is observed in rocks and materialssubjected to repeated cycles of loading and unloading where the

peak stress increases from cycle to cycle As long as the stressremains below the largest previously attained stress the AE countsremain at zero or close to the background level When reachingthe previous peak stress the system experiences a surge of AEactivity (Kurita amp Fujii 1979 Li amp Nordlund 1993 Pestman ampMunster 1996 Lavrov 2001) The Kaiser effect has been observedin a broad range of rocks and materials including granite (Kuritaamp Fujii 1979 Li amp Nordlund 1993) andesite (Yoshikawa amp Mogi1981) marble (Holcomb 1993a Li amp Nordlund 1993 Tuncay ampUlusay 2008) gneiss (Li amp Nordlund 1993) sandstone (Kaiser1953 Goodman 1963 Tuncay amp Ulusay 2008) limestone (Lavrov2001) and tuff (Tuncay amp Ulusay 2008) The quality of the Kaisereffect is characterized by the Felicity Ratio (FR) which is the ratioof the stress (σAE) at the onset of AE to the peak stress (σMax) of theprevious loading cycle In the ideal case FR = σAEσMax = 1 butin many cases it decreases during the experiment as shown in Fig 2with data from Li amp Nordlund (1993) When the stress during thecycle approaches the rock strength the Kaiser effect deterioratesand the FR steeply decreases This tendency is reproduced by themodel (coloured lines also shown in Fig 2) which is presentedbelow

The Kaiser effect is less pronounced when rocks are subjected tohigh shear deformation leading to dilation (Kurita amp Fujii 1979)The FR values are affected by changing the strain rate betweencycles (Lavrov 2001 Zhang et al 2018) as well as the temperature(Li amp Nordlund 1993) There have also been examples of sensitivityto healing processes (Lavrov 2003)

There is a general agreement that the Kaiser effect is connected todamage accumulation (eg Holcomb 1993b Lavrov 2003 Hamielet al 2004a) yet there is no general model explaining the difference

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Strain localization and Kaiser effect 2093

Figure 2 Felicity Ratio (FR) versus stress for three different types of rocks (marked with different coloured stars) obtained in the experiment of Li amp Nordlund(1993) Lines with the same colours represent simulated FR of this study

in the FR trends for various rock types as well as second-order ef-fects such as time and strain rate sensitivity A better understandingof the Kaiser effect and its sensitivities especially in relation to lo-calization processes may help to predict failure on different scales

13 Damage rheology modelling

Numerous experiments demonstrate that the yield surface of poroussandstones forms a cap-shaped line (eg Baud et al 2006 Tembeet al 2007) The lsquoyield caprsquo may be defined in notations of stressesor strains its shape along with the deformation path defines themode of failure (eg Baud et al 2004) Laboratory experiments ofporous rocks show evidence of overall strain hardening and yieldcap growth attributed to plasticity and porosity loss (eg Baud et al2006 Tembe et al 2007 Bedford et al 2018 2019) Several me-chanical lsquoyield caprsquo model formulations address the deformationalprocesses in porous rocks Elastoplastic formulation also known asbifurcation analysis (eg Rudnicki amp Rice 1975 Issen amp Rudnicki2000 Olsson amp Holcomb 2000 Schultz amp Siddharthan 2005) suc-cessfully predicts the orientation of deformation bands (sheer bands)but lacks any time dependency and temporal evolution of the bandsContinuum breakage models for mechanics of granular media ac-curately predict the yield surface and localization patterns of poroussandstones based on grain and cement properties (Das et al 2014Tengattini et al 2014) Grueschow amp Rudnicki (2005) discusseddifferent models of evolving yield cap (DiMaggio amp Sandler 1971Carroll 1991) and developed a constitutive model connecting theyield cap growth with different failure modes Results of numericalsimulations utilizing different formulations incorporating changesof the local elastic properties and porosity-induced yield cap changeexplicitly connecting different deformation patterns and modes offailure with the yield cap growth (eg Vorobiev et al 2007 Stefanovet al 2011 Lyakhovsky et al 2015)

The yield surface for crystalline and low-porosity rocks are oftenapproximated by a straight line representing the Coulomb failurecriterion yet the failure envelope of most rock types appears to

curve with growing compression and confining pressure (Jaegeret al 2007) Observed Kaiser effect in crystalline rock samplesindicates strain hardening in consecutive cycles and is attributedto the change in the yield surface due to damage accumulation(Holcomb 1993b Hamiel et al 2004a)

In order to connect between localized and distributed damage ac-cumulation during pre-faulting processes yield surface evolutionand the Kaiser effect we outline a new damage-rheology modelformulation with an evolving damage-dependent yield surface Weuse a semi-analytical approach and set of 1-D numerical simula-tions which explain damage accumulation patterns and connectsthem to the felicity ratio of the Kaiser effect We demonstrate thatthe model with a damage-dependent yield surface not only capableof reproducing the Kaiser effect but also connects the change in FRwith transitions between two modes of brittle deformation localizedversus distributed fracturing under low confining pressures

2 METHODS

21 Model description

In this study we follow the main derivations of the damage poro-elastic model by Hamiel et al (2004b) and Lyakhovsky et al (2015)considering two thermodynamic state variables porosity φ anddamage α coupled by kinetic relations The porosity refers to thevolume fraction of pores with complete unloading With this defini-tion the porosity can only change with inelastic deformation Thedamage ranges between 0 and 1 (α = 0 for intact material andα rarr 1 for total failure) and is responsible for the rock stiffness andits degradation induced by microcracks voids or any other typeof flaws The key thermodynamic relations are briefly discussed inthe Appendix The detailed derivations and numerical results canbe found in Lyakhovsky et al (2015) They demonstrated that atthe intermediate and high confining pressures competition betweeninelastic compaction (porosity decrease) and dilation (porosity in-crease) damage accumulation and healing leads to different modes

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Figure 3 Yield surface change in strain coordinates as a function of damageεv εd are volumetric and differential strain components respectively yieldcap growth parameter is D2 = minus3 and the power law is n = 2

of failure in porous rocks Incorporating porosity-dependent yieldsurface change allowed to successfully reproduce cataclastic flowwith no significant localization in porous rocks at high confiningpressures (above 170 MPa) and localized brittle failure at interme-diate confining pressures (below 90 MPa)

In this study we focus on the damage accumulation processes inthe compaction regime under uniaxial unconfined conditions andtherefore neglect the healing processes Also considering drainedconditions or dry rocks fluid pressure should be excluded Verysmall porosity changes have a minor effect on the stressndashstrainrelations and are also neglected The crucial modification of theLyakhovsky et al (2015) model is the assumption that the modelparameter controlling the size of the yield surface depends not onlyon the porosity but also on the damage variable Fig 3 shows aseries of yield surfaces for different damage values and one of thespecified sets of the model parameters connecting damage with theD-value in eq (4) (see the Appendix) A decrease in the D-valueleads to a self-similar increase of the yield surface

22 Model formulation

We consider 1-D uniaxial compression with all the model variablesdisplacement u(x t) strain ε(x t) = partupartx (negative strain valuesstand for compaction) and damage α(x t) depending only on thex coordinate and time t Under these conditions the 1-D damagekinetic equation (eq A5 from the Appendix) is

dα

dt= Cd

[D (α) (minusε)n+2 + ε2 (1 minus ξ0)

] (1)

where Cd is the damage rate coefficient power index n and functionD(α) control the shape and size of the yield surface The solutionof the 1-D equilibrium equation (partσpartx = 0) gives uniform stresseven for a heterogeneous material with damage-dependent Youngrsquosmodulus

E (x t) = E0 [1 minus α (x t)] (2)

where E0 is the Youngrsquos modulus of the intact material The linearrelation between Youngrsquos modulus and the damage is common formany damage rheology models and it is analogous to the com-plete 3-D model formulations We study damage evolution in a

sample with a unit length and fixed displacements at the bound-aries u(0 t) = 0 and u(1 t) = u0 Integrating the stressndashstrainrelation (σ = E(x t) middot ε(x t)) over the unit length of the sampleusing the displacement boundary conditions and eq (2) gives thetime-dependent stress

σ (t) = ε0 (t)

(3)

where = int10

dx(1minusα) and ε0 is the elastic strain applied to the

sample which is equal to u0 divided by the sample length Thestrain distribution is given by

ε (x t) = ε0 (t)

middot (1 minus α (x t)) (4)

Substituting (4) into (1) leads to the equation of the damageevolution for the investigated sample

dα

dt= Cdε

20[(1 minus α) ]minus2

times [D (α) (minusε0)n((1 minus α) )minusn minus (1 + ξ0)

] (5)

The evolution equation for the damage accumulation is only de-fined for conditions where the expression in the right square bracketsis positive For a non-uniform damage distribution it grows only inthe points where the strain values are above the yield surface Forconstant D-values the yield surface remains unchanged and dam-age keeps growing and strongly localizing until failure (Lyakhovskyet al 1997) Distributed strain and smoothing of the damage dis-tribution are possible for decreasing D-values which promote anincrease in the size of the yield surface Fast-growing yield surfacerelative to evolving load conditions may bring the yield path backto the stable conditions and thus smoothing the damage distribu-tion To realize such scenario a strong damage dependency of theD-values should be adopted Considering this condition we suggestan exponential D(α) dependency

D (α) = D1 exp (D2α) (6)

where positive coefficient D1 stands for the initial D-value of thedamage-free material and negative D2 controls its decrease withthe growing damage

Substituting (6) into the damage growth eq (5) allows to definea critical strain

εcr = minus(

1 + ξ0

D1

) 1n

(7)

corresponding to the onset of damage accumulation in the intactmaterial (α = 0) After scaling the overall strain and stress to thecritical values εlowast(t) = ε0εcr σlowast(t) = σE0εcr and using (6) thedamage kinetic eq (5) is re-arranged to a slightly simplified form

dα

dt= χε2

lowast(1 minus α)minus2minus2[exp (D2α) εnlowast ((1 minus α) )minusn minus 1

] (8)

where χ = Cd ε2cr(1 + ξ0) This form of the damage growth equa-

tion shows that the obtained damage pattern and distributed orlocalized strain depend only on two parameters that is applieddeformation εlowast and the coefficient D2 controlling the coupling be-tween damage and the size of the yield surface D1 is included inthe kinematic coefficient χ and does not affect the spatial damageevolution

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23 Modelling of the Kaiser effect

In this section we present an analytical estimation of the FR changeunder constant strain rate We assume that under low confiningpressures damage accumulation process consists mostly of fractureprocesses resulting from intergranular cracking breaking of ce-mented grain boundaries grain boundary sliding and grain rotationboundaries We assume that the damage parameter α correlates withthe cumulative AE counts (Hamiel et al 2004a Choens amp Chester2014) and that the onset of damage can be determined by the onsetof AE thus allowing us to infer the quality of the Kaiser effect andFR trend

We start with a simplified case of a homogeneous and initiallydamage-free sample (α(0) = 0) loaded at a constant strain rate εlowastOnce loading reaches the yield criterion damage starts to accumu-late and the yield criterion changes After a small amount of damageis accumulated the sample is unloaded and then reloaded at the samestrain rate as the previous cycle until yielding is achieved againThis new yield condition is reflected by the FR of the Kaiser ef-fect In the case of a homogeneous damage distribution α = α(t)the value in eq (4) becomes equal to (t) = 1(1 minus α(t)) and the strain distribution within the sample is also homogeneousεlowast(x t) = εlowast(t) Substituting the above value in eq (8) gives theonset of AE at the strain value equal to ε = exp(minus(D2n)α) andusing the corresponding stress the model FR is defined as

FR = σlowastAE

σlowastMax= (1 minus α) exp

(minus D2n α

)σlowastMax

(9)

where σlowastMax is the peak stress obtained during the first of twoconsecutive cycles During a loading cycle the stress continues togrow with increasing strain In some cases it reaches the ultimatevalue σlowastultimate and then slightly decreases before the unloading dueto a strong decrease in the effective Youngrsquos modules In most casessuch a decrease leads to a stress drop and macroscopic failure ofthe sample In our calculations we distinguish between these casesand use two different definitions for the maximum stress value

σlowastMaxsim=

(1 minus α) εlowast for Max (σlowast) lt σlowastultimate

σlowastultimate for Max (σlowast) = σlowastultimate (10)

The model with constant yield surface (D2 = 0) predicts thatthe FR decreases proportionally to the strain (FR = 1εlowast ) untilultimate stress is reached For a constant strain rate εlowast prop εlowastt the FRdecreases at the initial stage of loading with the slope equal to thestrain rate The analytical solution presented below demonstratesthat this tendency of the FR decrease is general for all negativeD2 values An analytical solution is obtained for the homogeneousdamage distribution for which the growth eq (8) is reduced to

dα

dt= χε2

lowast[exp (D2α) εnlowast minus 1

] (11)

We consider slow constant loading rates (εlowast = const) such thatthe damage closely follows the equilibrium damage value α0(t) =minus(nD2)ln(εlowast(t)) We find the solution of the eq (11) in the form ofsmall damage deviation (δα 1) from this equilibrium solution

α (t) = α0 (t) minus δα (t) = minus n

D2ln (εlowast (t)) minus δα (t) (12)

Keeping first-order terms for small δα leads to

d

dt(δα) = δα (t) D2χε2

lowast (t) + d

dt(α0 (t)) (13)

which has the solution

δα (t) = e f (t)

int t

0

dα0 (t)

dteminus f (t)dt (14)

where f (t) = D2χε2lowast(t) Integrating (14) substituting into (12) and

then into (9) leads to a semi-analytical approximation of the FR Ata very initial stage of loading (e f (t) = eminus f (t) asymp 1) δα can be ap-proximated as δα(t) sim (dα0(t)dt)t Finally substituting this valueinto (9) and using (10) gives the approximation for the initial stageof loading for negative D2 values

FR = 1 minus εlowastt (15)

3 RESULTS


The results of the analytical and numerical solutions are shown inFig 4 for two different strain rate values and different D2 valuesGreen lines represent the linear FR decrease at very initial stagesusing the approximation (15) This expression determines that for aconstant strain rate experiment the FR will always initially decreasewith a slope equal to εlowast The red lines in Fig 4 represent the semi-analytical solution by the integration of (14) Since α0(t) increaseslogarithmically in time and rate of its change decreases the solutionof (14) predicts that δα rarr 0 at a certain stage of growth givingminimum δα at a certain stage At this point depending on whetherσlowastultimate was reached the FR either starts increasing towards FR rarr1 (σlowastMax lt σlowastultimate) or reaches an inflection point and continuesdecreasing (σlowastMax = σlowastultimate ) FR increase is clearly recognizedin the case with D2 = minus3 shown in Fig 4(b) and less explicit ata lower strain rate shown in Fig 4(a) FR value linearly decreaseswith time at the very initial stage of loading then slightly increasesand finally decreases towards failure For largerD2 = minus2 value (alsoshown in Fig 4) the inflection point is clearly recognized Whenthe ultimate strength is reached so that σMax = σultimate FR becomeslinearly dependent on σlowastAE = (1 minus α) εlowast(t) and decreases towardszero as α rarr 1

The numerical code was developed for an explicit solution of thedamage growth eq (8) and for calculating the FR for any giveninitial damage distribution Numerical simulations and analyticalsolutions were done for many cases in which the time until unload-ing was changed and the new yield and damage was calculatedReloading the sample back to yield allowed the calculation of theFR for each unloading time Numerical results for the damage-freeinitial conditions (blue stars in Fig 4) match the analytical solutionincluding the initial decrease followed by a certain increase or aninflection point in the FR

In most studies dealing with the Kaiser effect the experimen-tally estimated FR is presented versus stress as shown in Fig 2Moreover in most rock mechanics experiments loading could beincreased up to the ultimate stress A negative slope of the stressndashstrain curve leads to instability and fast macroscopic failure occursif the loading system is not equipped by a sensitive servo-controlsystem (eg Lockner et al 1992) Since the stressndashstrain curve ofthe rock sample accumulating AE (damage) is nonlinear it is notpossible to make a direct comparison between simulated FR ofFig 4 with experimental data Therefore Fig 5 shows FR versusstress compatible with data reported by Li amp Nordlund (1993) tothe ultimate stress marked with stars at the end of each curve FR ismeasured from the onset of the damage accumulation at 20 MPa inthis model Similar to Fig 4 all the curves linearly decrease withthe onset of damage but soon after start to deviate according to themodel parameters The most pronounce FR decrease (black line inFig 5) corresponds to the model with n = 2 and D2 = minus4 as isexpected for larger negative D2 values Even for a very low damage

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Figure 4 The FR of two consecutive loading cycles as a function of the normalized unloading strain of the first cycle The results are for a homogeneousinitially intact sample (α(0) = 0) The simulated sample was loaded at a constant strain rate immediately unloaded at εlowast and then reloaded at the same rate untilα = 1 Blue stars show the current numerical resultsmdasheach star represent a single simulation The red line shows the results of the analytical approximation(see the text) The green line shows the initial linear dependency of the FR on the strain rate Panel (b) shows the results for a strain rate 10 times larger thanthat given in panel (a)

Figure 5 FR versus stress for different model realizations with power index n = 12 D2 = minus4 constant and stress-dependent damage rate coefficient

accumulation rate (damage rate is three orders of magnitudes belowthe strain rate) the ultimate stress is well below the strength of therock samples of Fig 2 (70 MPa versus 130 MPa) A larger ultimatestress value is predicted by a model with lower value of the powerindex n = 1 With the reduced ratio between damage rate to thestrain rate to 01 the ultimate stress reaches about 100 MPa (greenline in Fig 5) and FR decrease is less pronounced In these two

cases the damage rate coefficient is constant and the shape of bothgreen and dark blue curves is far from the general tendency of theFR data of Fig 2

Comparing measured and predicted strength of granite samplesunder various loading rates and confining pressures Lyakhovskyet al (2005) suggested that experimental observations are fittedwell with constant damage rate coefficientCd under high confining

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pressure At low confining pressures the damage rheology modelwith constant Cd values significantly underestimated the rate ofdamage accumulation and overestimated the rock strength There-fore they suggested exponentially stress-dependent Cd whichstrongly decreases with increasing load

Cd (σ ) = Cd0 + Cd1 exp (σσ0) (16)

For Westerly granite a stress-dependent Cd decreases with a char-acteristic scale of σ0 = 50 MPa (Lyakhovsky et al 2005) Asimilar exponential decrease in Cd used by Panteleev et al (2020)who analysed damage accumulation in Darley Dale sandstone un-der 4 MPa confining pressure With this stress dependent Cd thedecrease of FR (blue line in Fig 5) is much smaller with no changeof the strength (compare with green line) This model was runwith Cd0 = Cd1 equal to the strain rate and σ0 = 40 MPa Asignificant increase of Cd1 by two orders of magnitudes further re-duces FR decrease during the initial stage of loading (red line inFig 5) and a similar decrease of Cd0 by two orders of magnitudesleads to gradual FR decrease at large stresses The shape of thered line Fig 5 is close to the FR change of the Bjorka marble dataof Fig 2

The simulated FR curves of Fig 2 are achieved with an additionaltuning of the model parameters The curves reasonably reproducethe general FR tendency for Bjorka marble Malmberget and Bol-men gneiss For all cases the power index n is 1 D2 = minus38 forBjorka marble and Malmberget gneiss andD2 = minus45 used for Bol-men gneiss The characteristic stress value σ0 = 45 16 and 22MPa and the ratio Cd1Cd0 is 106 103 and 105 for Bjorka mar-ble Malmberget and Bolmen gneiss respectively We note here anon-uniqueness of the model parameters with the lack of detailedinformation on the experimental settings and additional experi-ments results under different confined conditions The presentedmodel results in Figs 2 and 5 demonstrate the general ability ofthe model in reproducing the Kaiser effect and tendency in theFR change

32 Distributed versus localized damage accumulation

We utilize the same numerical code to study the evolving spatialdistribution of the damage and strain in the sample To distinguishbetween effects associated with the rate of damage growth and therate of loading we apply an instantaneous load and then fixed dis-placements at the edges of the sample The rate of the strain anddamage change in the sample is controlled by a single model pa-rameter χ prop CdD1 (damage kinetic coefficient) Therefore all theresults are shown versus non-dimensional time tlowast = t middot (CdD1)In this case the results depend on the applied strain (εlowast) damage-dependent change on the yield surface (D2) and the initial dam-age distribution (α0(x)) In the case of the constant yield surface(D2 = 0) singular damage and strain localization is obtained (notshown here) similarly to the local damage rheology models (egLyakhovsky et al 1997) However an evolving yield with D2 lt 0may lead to a stabilization of the damage growth and prevent strainlocalization We demonstrate this behaviour for D2 = minus3 using asimplified initial box-like damage distribution with a small area of10 of the sample length with initial damage level α2 = 03 whichdiffers from the rest of the sample with a damage level α1 = 02 Thebox-like spatial damage distribution is preserved during evolution(Figs 6a and b) while the difference between the evolving dam-age values α(t) = α2(t) minus α1(t) determines the tendency of thesystem to exhibit localized (dαdt gt 0) or distributed

(dαdt lt 0) damage distribution Under relatively high strain( εlowast = 2) the sample undergoes damage localization damagewithin the initially weak area increases much faster than its sur-roundings (Fig 6a) and α steeply increases with time towardstotal failure (Fig 6c) Under lower strain ( εlowast = 14) only veryminor damage growth is predicted in the initially weak zone whilemost of the growth occurs in the surrounding material (Fig 6b) Thedifference between damage values α decreases towards a con-stant value (Fig 6d) These results mean that under different loadingconditions the same material can undergo strain and damage local-ization and macroscopic failure or distributed damage accumulationwith a steady-state condition

Fig 7 shows the result of a parametric search study for differentloading conditions yielding parameters and initial damage distri-bution Two different cases are considered In one case α2 ge α1

representing a small embedded weak zone as in Fig 6 Simula-tions with this initial condition form the upper triangle part of eachdiagram The lower triangle corresponds to the strong embeddeddamage zone with α2 lt α1 The coloured fields of each diagramcorrespond to different trends of damage evolution Yellow fieldscorrespond to the stable condition below the yield surface The sta-ble area occupies almost the whole diagram (c) corresponding toa relatively high D2 = minus4 and small loading εlowast = 12 since largenegative D2 values are a stabilizing factor and low strains lead toless pronounced damage Making D2 values less negative alongwith load increase leads to shrinking the stable zone increasing theblue area where localized damage is predicted In diagram (g) theblue zone occupies significant part of the whole area Arrows insidethis zone show the general tendency of the damage evolution Oncelocalization conditions for weak inclusion (upper triangle of thediagrams) are realized damage evolves towards failure while forthe strong inclusion (lower triangle) the background (α1) damagevalue increases until the stable state is reached Red zones appearin some diagrams pointing to the conditions of the distributed dam-age eliminating spatial damage variations In this regime damageevolves towards a stable state With the same model parameters (ieD2 value) the geometry of these zones is very sensitive to the ap-plied load The model with D2 = minus3 and the same initial damagedistribution predicts distributed damage at the intermediate strainlevel [green star in diagram (e)] and localized damage at the el-evated strain [red star in diagram (h)] The damage evolution forthese cases is shown in the previous Fig 6

An additional series of simulations with initial random damagedistribution α(0) = 02 plusmn 002 strain εlowast = 18 and different D2

values (Fig 8) shows the effect the small change in the D2 val-ues from minus30 to minus325 on the mode of the damage evolutionFor D2 = minus30 (Fig 8a) the damage growth significantly acceler-ates in the zones with elevated initial damage values finally form-ing a spike around x sim 03 Similar to Figs 6(c) and (d) we useα = αMax minus αMin as a measure of the spatial damage heterogene-ity α slowly increases during most of the loading period (Fig 8b)but steeply increases in an explosive manner just prior to failure Theopposite behaviour is predicted by the model with slightly more neg-ative D2 = minus325 (Figs 8c and d) The same initial damage growswith an almost similar rate in the whole sample and is stabilized atvalues α lt 04 (Fig 8c) The spatial damage variability (Fig 8d)slightly increases at the initial stage of evolution but then grad-ually decreases (α rarr 0) These results obtained with the sameinitial and loading conditions show the transition between differentmodes of damage accumulation due to small variations in the singlemodel parameter D2 controlling the coupling damage-dependentyield surface change

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Figure 6 Time evolution of the spatial damage distribution (a and b) and the damage contrast α (c and d) in a sample with a bimodal damage distributionThe initial damage value of 90 of the sample is α1 = 02 the remaining 10 of the sample is assigned with the initial value of α2 = 03 The area with α2

damage value can be viewed as a damaged zone The simulations are conducted with the same D2 = minus3 and two different strain values εlowast = 2 (a and c) andεlowast = 14 (b and d)

33 The Kaiser effect and damage localization

We study the connection between the modes of damage evolu-tion distributed versus localized and the Kaiser effect quality ex-pressed by FR The results shown in Fig 9 are obtained using thesame numerical procedure with random initial damage distribu-tion α(0) = 02 plusmn 002 but applying a constant strain rate Asin previous simulations (Fig 4) the FR is calculated using twoconsecutive loading cycles and is presented as a function of thenormalized unloading time (blue line in Fig 9) Due to relativelyhigh D2 = minus345 and low strain at the initial stage of loading FRremains high during a relatively long stage until about εlowast sim 17 Atthe following stage the decreases of the FR value slowly acceleratesand drops below FR = 09 at the final stage towards failure Thered line on the same plot shows the measure of the spatial damageheterogeneity α = αMax minus αMin Similarly to the previous results(Figs 6 and 8) this difference points to transition from a distributeddamage accumulation mode with decreasing spatial damage vari-ability to the localization mode The clear minimum (star on thered line) shows that this transition occurs around εlowast sim 17 close tothe onset of the FR decrease The comparison between blue and redlines shown on the same plot in Fig 9 suggests that high FR valuesare expected during the loading stage with a distributed mode ofthe damage accumulation This and other simulation results underdifferent conditions (not shown here) support that for highly dam-age dependent yield surfaces high FR values are expected duringthe loading stage corresponding to the distributed damage accu-mulation mode The damage and strain localization occurs underconditions corresponding to a steep FR decrease

4 D ISCUSS ION

The presented results highlight the coupling between the damageand the yield surface growth and show its effect on damage localiza-tion patterns as well as the quality of the Kaiser effect as reflectedby the FR Weak coupling between the damage and yield surfacegrowth results in an immediate decrease of the FR A stronglycoupled damagemdashyield surface growth result with an initial de-crease of the FR and then an increase to a value close to 1 untilapproaching the samplesrsquo ultimate strength When the sample ap-proaches failure FR rarr 0 The change in the FR has been commonlynoted in laboratory experiments yet is rarely quantified (Zhang et al2018) Li amp Nordlund (1993) presented results demonstrating FRvalues for three different types of rocks (Fig 2) Coloured lines inFig 2 show simulations of FR various stress Reasonable agree-ment between the general tendency of the experimentally observedchange of the FR values with loading and modelling results pro-vides validating the developed model Comparing the model resultsobtained with constant and stress-dependent damage rate coeffi-cient (Fig 5) emphasizes its significant increase under low-stressvalues

Localization patterns are connected to the coupling between yieldsurface and damage represented here by D2 Crystalline rocks thatshow strong localization patterns are expected to have D2 valuesclose to zero A weak coupling between damage and yield surfacedoes not allow strain hardening thus damage continues to accu-mulate in the most damaged areas promoting unstable localizedfracturing Rocks with damage-dependent yield surface controlledby negative D2 values exhibit a significant distributed damage stage

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 7 A set of diagrams demonstrating different modes of damage evolution The yellow areas mean stable damage distribution the blue areas meandamage localization and the red areas mean distributed damage eliminating spatial damage variations Each diagram represents simulations with bimodalinitial damage distribution with α1 and α2 values similar to Fig 6 The simulations of Fig 6 are marked by red and green stares Different panels correspondto different applied constant strain εlowast and yield cap growth parameter D2 values

Every column in Fig 6 can be viewed as snapshots of a continuouslaboratory experiment with increasing loading Similar to labora-tory experiments the model predicts that initial loading starts withstable conditions (yellow) and with increasing loading the samplegets into the mode of distributed damage (red) and with furtherloading becomes localized (blue) For example the green and redstars in Figs 6(e) and (h) represent the transition from a distributedto a localized mode of damage Two types of modelling cyclicloading for Kaiser effect and continuous loading for localizationshown in Fig 9 demonstrate that the transition between distributedand localized modes of damage is correlated with the onset of asteep decrease in FR

Some works have experimentally examined the overall yield sur-face evolution of porous rocks (Bedford et al 2018 2019) but theconnection between the rocksrsquo physical properties and the shape of

the yield surface is not clear The presented model allows a con-straint of the yield surface parameters by monitoring AE duringcyclic load and estimating FR The ratio D2n can be estimated byrearranging (9)

D2

n= minus E0

E0 minus Eeffln

(FR σMax

Eeffεcr

)(17)

Using this relation the D2n ratio can be estimated from the FRvalue and the effective Youngrsquos modulus from measured stressndashstrain curves D1 and n can be independently constrained by deter-mining the shape of the yield surface for an intact rock by a seriesof loading experiments under different confining pressures

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Figure 8 Time evolution of the spatial damage distribution (a and b) and the damage contrast α (c and d) in a sample with initial random damage distributionα(0) = 02 plusmn 002 constant applied strain εlowast = 18 and a slightly different yield cap growth parameter D2 = minus30 (a and b) and D2 = minus325 (c and d) Thesimulation shows that the transition from localized failure (a and b) to the distributed damage occurs at small variations of the D2 value The difference betweenthe maximum and minimum damage α = αMax minus αMin steeply increases towards failure (b) while for more negative D2 (d) after some small increase atthe initial stage α gradually decreases

Figure 9 Simulated FR (the blue line) and the measure of the spatial damage heterogeneity (α = αMax minus αMinmdash red line) for a model with initial randomdamage distribution α0 = 02 plusmn 002 yield cap growth parameter D2 = minus345 and constant strain rate The white star and the dashed black line mark the FRlocal maxima that coincides with the transition of the system from the distributed mode of the damage accumulation to localization

Although mostly observed under controlled conditions in thelab experiments the Kaiser effect has been observed in dif-ferent scales and settings It was tested for practical use as apossible indicator for meters long concrete beam failure (Sagaret al 2015) and was possibly observed during AE monitoring ofbridges (Nair amp Cai 2010) The effect was also observed in sev-eral locations on a kilometre-scale geological setting as seismicity

induced by pressure change due to volcanic activity (Heimissonet al 2015) and groundwater level changes (Simpson amp Negmat-ullaev 1981) Fault localization processes have been observed ondifferent scales (Ben-Zion amp Sammis 2003) including tectonic set-tings (eg Marco 2007) However there is an ongoing discussion re-garding the similarity of these process on different scales (Ben-Zionamp Sammis 2003) Our formulation is dimensionless suggesting that

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the connection between the FR and localization process may be ob-served at all scales from micro fractures to plate-boundary faultingthus in cases where highly negative D2 values are matched to aproblem changes in the FR trend may help to predict damage lo-calization process and failure

5 CONCLUS IONS

We study the effect of damage dependent yield surface on local-izeddistributed damage accumulation and the Kaiser effect usinganalytical and numerical methods Under sufficient strain a con-stant or weakly coupled damagemdashyield surface growth results instrong and fast damage-localization and failuremdashas seen in crys-talline rocks A strong coupling between damage and yield surfacegrowth promotes under certain conditions distributed damage ac-cumulation and for adequate strain paths induces transition be-tween phases of localized and distributed damage accumulationThis type of dependency also allows the material to regain stabil-ity after initial damage increase and can promote the consecutiveformation of deformation bands prior to faulting

The quality of the Kaiser effect produced by rock samples con-secutive loading cycles is determined by the FR While for weakdamagemdashyield growth coupling the FR deteriorates between con-secutive cycles for a strong coupling the FR remains close to oneand starts dropping upon approaching ultimate strength and de-teriorates towards zero as the sample reaches failure For strongdamagemdashyield surface coupling the FR decrease correlates withthe transition between distributed and localized damage accumula-tion We suggest that a better understanding of FR trends can beused to determine the yield surface dependency on damage andhelps predict different damage localization patterns

ACKNOWLEDGEMENTS

The paper was benefitted from constructive comments by JessMcBeck anonymous reviewer and the Editor Alexis Maineult Thestudy was supported by grant from the United States-Israel Bina-tional Science Foundation (BSF 2014036) WZ acknowledges thepartial support from NSF EAR-1761912 SM acknowledges supportof the Israel Sinence Foundation 164519

REFERENCESAntonellini AM Aydin A amp Pollard D 1994 Microstructure of de-

formation bands in porous sandstones at Arches National Park Utah JStruct Geol 16(7) 941ndash959

Aydin A amp Johnson AM 1983 Analysis of faulting in porous sandstonesJ Struct Geol 5(1) 19ndash31

Baud P Klein E amp Wong T 2004 Compaction localizetion in poroussandstones spatial evolution of damage and acoustic emission activity JStruct Geo 26603ndash624

Baud P Vajdova V amp Wong T 2006 Shear-enhanced com-paction and strain localization Inelastic deformation and constitu-tive modeling of four porous sandstones J geophys Res 111(B12)doi1010292005JB004101

Bedford JD Faulkner DR Leclere H amp Wheeler J 2018 High-resolution mapping of yield curve shape and evolution for porous rockthe effect of inelastic compaction on porous bassanite J geophys Res123(2) 1217ndash1234

Bedford JD Faulkner DR Wheeler J amp Leclere H 2019 High-resolution mapping of yield curve shape and evolution for high porositysandstone J geophys Res 124(6) 5450ndash5468

Ben-Zion Y amp Sammis CG 2003 Characterization of fault zones Pureappl Geophys 160(3-4) 677ndash715

Berdichevsky VL 2009 Variational Principles of Continuum MechanicsSpringer-Verlag

Bernard X Du Labaume P Darcel C Davy P amp Bour O 2002 Cata-clastic slip band distribution in normal fault damage zones Nubian sand-stones Suez rift J geophys Res 107(B7) doi1010292001JB000493

Biot MA 1955 Theory of elasticity and consolidation for a porousanisotropic solid J Appl Phys 26(2) 182ndash185

Carroll MM 1991 A critical state plasticity theory for porous reservoirrock in Recent Advances in Mechanics of Structured Continua Vol117 pp 1ndash8 eds Massoudi M Rajagopal KR American Society ofMechanical Engineers

Choens RC amp Chester FM 2014 Characterizing damage evolution andyield surfaces for berea sandstone under triaxial loading as a functionof effective pressure in 48th US Rock MechanicsGeomechanics Sympo-sium American Rock Mechanics Association

Das A Tengattini A Nguyen GD Viggiani G Hall SA amp EinavI 2014 A thermomechanical constitutive model for cemented granu-lar materials with quantifiable internal variables Part IIndashvalidation andlocalization analysis J Mech Phys Solids 70 382ndash405

de Groot SR amp Mazur P 1962 Nonequilibrium Thermodynamics pp510 North-Holland Publishing Co

DiMaggio FL amp Sandler IS 1971 Material model for granular soils JEng Mech 97(EM3) 935ndash950

Fitts DD 1962 Nonequilibrium Thermodynamics 173 pp McGraw-HillBook Company

Fossen H Schultz RA Shipton ZK amp Mair K 2007 Deformationbands in sandstone a review J geol Soc Lond 164(4) 755ndash769

Gajst H Weinberger R Zhu W Lyakhovsky V Marco S amp Shalev E2018 Effects of pre-existing faults on compaction localization in poroussandstones Tectonophysics 747 1ndash15

Goodman RE 1963 Sub audible noise during compression of rocks Bullgeol Soc Am 74(4) 487ndash490

Grueschow E amp Rudnicki JW 2005 Elliptic yield cap constitutive mod-eling for high porosity sandstone Int J Solids Struct 42(16ndash17) 4574ndash4587

Hamiel Y Liu Y Lyakhovsky V Ben-Zion Y amp Lockner D 2004aA viscoelastic damage model with applications to stable and unstablefracturing Geophys J Int 159(2004) 1155ndash1165

Hamiel Y Lyakhovsky V amp Agnon A 2004b Coupled evolution ofdamage and porosity in poroelastic media theory and applications todeformation of porous rocks Geophys J Int 156(3) 701ndash713

Heimisson ER Einarsson P Sigmundsson F amp Brandsdottir B 2015Kilometer-scale Kaiser effect identified in Krafla volcano Iceland Geo-phys Res 42(19) 7958ndash7965

Holcomb DJ 1993a Observations of the Kaiser effect under multiaxialstress states implications for its use in determining in situ stressGeophysRes 20(19) 2119ndash2122

Holcomb DJ 1993b General theory of the Kaiser effect Int J RockMechMin Sci Geomech Abstr 30(7) 929ndash935

Issen KA amp Rudnicki JW 2000 Conditions for compaction bands inporous rock J geophys Res 105(B9) 21 529ndash21 536

Jaeger JC Cook NGW amp Zimmerman R 2007 Fundamentals of RockMechanics John Wiley amp Sons

Kaiser J 1953 Erkenntnisse und Folgerungen aus der Messung vonGerauschen bei Zugbeanspruchung von metallischen Werkstoffen ArchEisenhuettenwes 24(1-2) 43ndash45

Kurita K amp Fujii N 1979 Stress memory of crystalline rocks in acousticemission Geophys Res 6(1) 9ndash12

Lavrov A 2001 Kaiser effect observation in brittle rock cyclically loadedwith different loading rates Mech Mater 33(11) 669ndash677

Lavrov A 2003 The Kaiser effect in rocks principles and stress estimationtechniques Int J Rock Mech Min Sci 40(2) 151ndash171

Li C amp Nordlund E 1993 Experimental verification of the Kaiser effectin rocks Rock Mech Rock Eng 26(4) 333ndash351

Lockner DA Byerlee JD Kuksenko V Ponomarev A amp SidorinA1992 Observations of quasistatic fault growth from acoustic

Dow




ay 2020

2102 HGajst et al

emissions in Fault Mechanics and Transport Properties of Rocks Vol51 pp 3ndash31 eds Evans B amp Wong T Academic Press San DiegoCalif

Lyakhovsky V Ben-Zion Y amp Agnon A 1997 Distributed damage fault-ing and friction J geophys Res 102(B12) 27 635ndash27 649

Lyakhovsky V Ben-Zion Y amp Agnon A 2005 A viscoelastic damagerheology and rate- and state-dependent friction Geophys J Int 161179ndash190

Lyakhovsky V Zhu W amp Shalev E 2015 Visco-poroelastic damagemodel for brittle-ductile failure of porous rocks J geophys Res 120(4)2179ndash2199

Mair K Main I amp Elphick S 2000 Sequential growth of deformationbands in the laboratory J Struct Geol 22(1) 25ndash42

Malvern LE 1969 Introduction to theMechanics of aContinuumMediumpp 713 Prentice-Hall Inc

Marco S 2007 Temporal variation in the geometry of a strikendashslip faultzone examples from the Dead Sea Transform Tectonophysics 445(3-4)186ndash199

Martyushev L amp Seleznev V 2006 Maximum entropy production principlein physics chemistry and biology Phys Rep 426 1ndash45

Nair A amp Cai CS 2010 Acoustic emission monitoring of bridges reviewand case studies Eng Struct 32(6) 1704ndash1714

Olsson WA amp Holcomb DJ 2000 Compaction localization in porousrock Geophys Res 27(21) 3537ndash3540

Onsager L 1931 Reciprocal relations in irreversible processes Phys Rev37 405ndash416

Panteleev I Lyakhovsky V Browning J Meredith PG Healy D ampMitchell TM 2020 Non-linear anisotropic damage rheology modeltheory and experimental verification J Mech Phys Solids submitted

Pestman BJ amp Van Munster JG 1996 An acoustic emission study ofdamage development and stress-memory effects in sandstone Int J RockMech Min Sci Geomech Abstr 33(6) 585ndash593

Prigogine I 1955 Introduction to Thermodynamics of Irreversible Pro-cesses Springfield

Rudnicki JW amp Rice JR 1975 Conditions for the localization of de-formation in pressure-sensitive dilatant materials J Mech Phys Solids23(6) 371ndash394

Sagar RV Prasad BR amp Singh RK 2015 Kaiser effect observation inreinforced concrete structures and its use for damage assessment ArchCiv Mech Eng 15(2) 548ndash557

Schultz RA amp Siddharthan R 2005 A general framework for the oc-currence and faulting of deformation bands in porous granular rocksTectonophysics 411(1-4) 1ndash18

Sedov LI 1968 Variational methods of constructing models of continuousmedia in Irreversible Aspects of Continuum Mechanics and Transfer ofPhysical Characteristics in Moving Fluids eds Parkus H amp Sedov LIpp 17ndash40 Springer-Verlag

Shipton ZK amp Cowie PA 2001 Damage zone and slip-surface evolutionover μm to km scales in high-porosity Navajo sandstone Utah J StructGeol 23(12) 1825ndash1844

Simpson DW amp Negmatullaev SK 1981 Induced seismicity at Nurekreservoir Tadjikistan USSR Bull seism Soc Am 71(5) 1561ndash1586

Stefanov YuP Chertov MA Aidagulov GR amp Myasnikov AV 2011Dynamics of inelastic deformation of porous rocks and formation oflocalized compaction zones studied by numerical modeling J MechPhys Solids 59(11) 2323ndash2340

Tembe S Vajdova V Baud P Zhu W amp Wong T 2007 A new method-ology to delineate the compactive yield cap of two porous sandstonesunder undrained condition Mech Mater 39(5) 513ndash523

Tengattini A Das A Nguyen GD Viggiani G Hall SA amp EinavI 2014 A thermomechanical constitutive model for cemented granularmaterials with quantifiable internal variables Part ImdashTheory J MechPhys Solids 70 281ndash296

Truesdell C amp Noll W 2004 The Non-Linear Field Theories of Mechan-ics 3rd edn pp 627 ed Antman SS Springer-Verlag

Tuncay E amp Ulusay R 2008 Relation between Kaiser effect levels andpre-stresses applied in the laboratory Int J Rock Mech Min Sci 45(4)524ndash537

Vorobiev OYu Liu BT Lomov IN amp Antoun TH 2007 Simulationof penetration into porous geologic media Int J Impact Eng 34(4)721ndash731

Weinberger R Baer G Shamir G amp Agnon A 1995 Deformation bandsassociated with dyke propagation in porous sandstone Makhtesh RamonIsrael in The Physics and Chemistry of Dykes eds Baer G amp HeimannA pp 95ndash112 Balkema

Yoshikawa S amp Mogi K 1981 A new method for estimation of the crustalstress from cored rock samples laboratory study in the case of uniaxialcompression Tectonophysics 74(3-4) 323ndash339

Zhang M Meng Q Liu S Qian D amp Zhang N 2018 Impactsof cyclic loading and unloading rates on acoustic emission evolu-tion and felicity effect of instable rock mass Adv Mater Sci Engdoi10115520188365396

APPENDIX

To account for the coupled evolution of damage and porosity inporoelastic media we generalize the results of Hamiel et al (2004b)and Lyakhovsky et al (2015) during irreversible brittle deforma-tion We follow the approach of irreversible thermodynamics (egOnsager 1931 Biot 1955 Prigogine 1955 Truesdell amp Noll 2004)which was successfully applied to kinetics of chemical reactionsand phase transitions (eg de Groot amp Mazur 1962 Fitts 1962)and served as a basis for variational methods of continuous mediamodels (eg Sedov 1968 Malvern 1969 Berdichevsky 2009) Theconstitutive behaviour of the material and flow rules controlling thekinetics of related irreversible processes are entirely defined by thespecification of two potentials free energy and dissipation func-tion or local entropy production Using the balance equations of theenergy and entropy Gibbs relation fluid mass conservation equa-tion the definitions of the stress tensor and fluid pressure Hamielet al (2004b) derived the local entropy production 13 related to thekinetics of porosity φ and damage α

13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)

where Pe is the effective pressure or mean stress minus fluid pres-sure and F is the free energy Onsager (1931) theoretically gener-alized the empirical laws of Fourier Ohm Fick and Navier repre-senting the specific local entropy production as a product of ther-modynamic fluxes and thermodynamic forces and adopting linearrelations between them (see review by Martyushev amp Seleznev2006) Following the Onsager principle Hamiel et al (2004b) andLyakhovsky et al (2015) formulated the kinetic relations for dam-age and porosity evolution as a system of equation

dφ

dt= minusCφφ Pe minus Cφα

partF

partα(A2)

dα

dt= minusCαφPe minus Cαα

partF

partα (A3)

where Cαα Cφφ Cαφ Cφα are kinetic coefficients or functions ofthe state variables The diagonal terms first in (A2) and secondin (A3) are responsible for the evolution of α and φ as functionsof damage and porosity related forces while the other terms areresponsible for the coupling between the kinetics of damage andporosity Hamiel et al (2004b) suggested that Cαφ is a power-lawexpression of the effective pressure Cαφ simPe

n They demonstratedthat the transition from positive to negative values of the slopeof the yield curve (yield cap) is a general feature of the modelWith this form of the kinetic coefficient their model defines theyielding condition in stress-space as being a function of the effective

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pressure Their formulation actually mixed stressndashspace and strainndashspace formulation for the kinetics of damage Lyakhovsky et al(2015) fixed this shortcoming formulation converting stress intostrainndashspace relations (see their eq 14)

dα

dt= Cd

[D (α ϕ) (minusI1)n+2 + I2 (ξ minus ξ0)

](A4)

where I1 = εkk I2 = εi j εi j are invariants of the strain ten-sor ξ = I1

radicI2 is the strain invariant ratio and its critical value

ξ = ξ0 controlling dilation-compaction transition The deriva-tive partFpartα sim minusI2(ξ minus ξ0) gives the second term in (A4) in-creasing with strain in power two while coupling the term withCαφ sim (minusI1)n+1 with n gt 0 increases with strain in a higher powerand becomes dominant at elevated strain values The competitionbetween these terms leads to the damage increase even below thedilation-compaction transition at ξ = ξ0 (see Lyakhovsky et al2015 for details) In such a regime with ξ lt ξ0 the damage-relatedterm in (A1) is negative which seems to violate the non-negativedissipation However even small compaction (dφdt lt 0) fixes this

problem All or part of the energy release due to porosity decrease istransferred into the energy needed for fracturing providing positivedissipation When the loading conditions overcome the dilation-compaction condition (ξ gt ξ0) the damage related dissipation be-comes positive and the energy goes into dilation In this study weare focused only on the details of the damage growth without goinginto details of the porosity change For the drained conditions ordry rocks with constant or zero fluid pressure it does not affect thestress-strain relations and could be excluded from the considerationFor the 1-D problem considered in this study addressing uniaxialcompression the strain invariants are I1 = minus radic

I2 = ε(x t) and thestrain invariant ratio is ξ = minus1 which is below typical ξ0-valuesestimated by Lyakhovsky et al (1997) from the internal frictionangle Substituting these values into (A4) leads to the simplifiedequation of the damage evolution

dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

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2092 HGajst et al

Figure 1 A schematic drawing of the damage accumulation stages in crystalline rocks and porous sandstones Crystalline rocks show strong damagelocalization while porous sandstone may experience several localization cycles in the form of multiple deformation bands forming prior to the developmentof a slip surface (ie fault)

zone of shear bands could continue to grow even after the forma-tion of a slip surface thus exhibiting distributed strain accumulationafter faulting

In the laboratory acoustic emission (AE) is measured to probemicrocracking during brittle deformation Lockner et al (1992)recorded locations of AE activities during fault propagation in ax-ially loaded Westerly granite samples Their results show that afteran initial lsquoringing phasersquo where AE was recorded throughout thesample a fast localization process occurred This coincided withthe development of a localized fault zone and failure In the samestudy when subjected to similar experimental conditions Webersandstone samples underwent a slow transition from tabular dis-tributed zones of AE activity to a localized planer zone of AEactivity (Lockner et al 1992) The sandstone samples did not ex-hibit an initial ringing phase and their final deformation zones werethicker than those of the granite samples Results of experiments ofLocharbriggs sandstone (Mair et al 2000) demonstrated the gradualformation of a package of shear bands clearly relating the amount ofdeformation and the number of strands (ie bands) They suggestedthat each strand accumulates a finite amount of strain before a newstrand is formed This can be interpreted as multiple transitionsbetween modes of localized and distributed damage accumulationThe transition mechanism is difficult to explain since no evidenceof strain hardening was observed in this experiment


The Kaiser effect (Kaiser 1953) is observed in rocks and materialssubjected to repeated cycles of loading and unloading where the

peak stress increases from cycle to cycle As long as the stressremains below the largest previously attained stress the AE countsremain at zero or close to the background level When reachingthe previous peak stress the system experiences a surge of AEactivity (Kurita amp Fujii 1979 Li amp Nordlund 1993 Pestman ampMunster 1996 Lavrov 2001) The Kaiser effect has been observedin a broad range of rocks and materials including granite (Kuritaamp Fujii 1979 Li amp Nordlund 1993) andesite (Yoshikawa amp Mogi1981) marble (Holcomb 1993a Li amp Nordlund 1993 Tuncay ampUlusay 2008) gneiss (Li amp Nordlund 1993) sandstone (Kaiser1953 Goodman 1963 Tuncay amp Ulusay 2008) limestone (Lavrov2001) and tuff (Tuncay amp Ulusay 2008) The quality of the Kaisereffect is characterized by the Felicity Ratio (FR) which is the ratioof the stress (σAE) at the onset of AE to the peak stress (σMax) of theprevious loading cycle In the ideal case FR = σAEσMax = 1 butin many cases it decreases during the experiment as shown in Fig 2with data from Li amp Nordlund (1993) When the stress during thecycle approaches the rock strength the Kaiser effect deterioratesand the FR steeply decreases This tendency is reproduced by themodel (coloured lines also shown in Fig 2) which is presentedbelow

The Kaiser effect is less pronounced when rocks are subjected tohigh shear deformation leading to dilation (Kurita amp Fujii 1979)The FR values are affected by changing the strain rate betweencycles (Lavrov 2001 Zhang et al 2018) as well as the temperature(Li amp Nordlund 1993) There have also been examples of sensitivityto healing processes (Lavrov 2003)

There is a general agreement that the Kaiser effect is connected todamage accumulation (eg Holcomb 1993b Lavrov 2003 Hamielet al 2004a) yet there is no general model explaining the difference

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2 METHODS



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dα

dt= Cd


] (1)


E (x t) = E0 [1 minus α (x t)] (2)



σ (t) = ε0 (t)

(3)

where = int10



ε (x t) = ε0 (t)



dα

dt= Cdε



] (5)


D (α) = D1 exp (D2α) (6)



εcr = minus(

1 + ξ0

D1

) 1n

(7)


dα

dt= χε2


] (8)



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FR = σlowastAE


(minus D2n α

)σlowastMax

(9)


σlowastMaxsim=




dα

dt= χε2


] (11)





d


lowast (t) + d

dt(α0 (t)) (13)


δα (t) = e f (t)

int t

0

dα0 (t)





3 RESULTS





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Cd (σ ) = Cd0 + Cd1 exp (σσ0) (16)










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4 D ISCUSS ION



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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)





D2

n= minus E0

E0 minus Eeffln

(FR σMax

Eeffεcr

)(17)


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5 CONCLUS IONS



ACKNOWLEDGEMENTS




































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APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



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dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

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2 METHODS



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dα

dt= Cd


] (1)


E (x t) = E0 [1 minus α (x t)] (2)



σ (t) = ε0 (t)

(3)

where = int10



ε (x t) = ε0 (t)



dα

dt= Cdε



] (5)


D (α) = D1 exp (D2α) (6)



εcr = minus(

1 + ξ0

D1

) 1n

(7)


dα

dt= χε2


] (8)



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FR = σlowastAE


(minus D2n α

)σlowastMax

(9)


σlowastMaxsim=




dα

dt= χε2


] (11)





d


lowast (t) + d

dt(α0 (t)) (13)


δα (t) = e f (t)

int t

0

dα0 (t)





3 RESULTS





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Cd (σ ) = Cd0 + Cd1 exp (σσ0) (16)










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4 D ISCUSS ION



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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)





D2

n= minus E0

E0 minus Eeffln

(FR σMax

Eeffεcr

)(17)


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5 CONCLUS IONS



ACKNOWLEDGEMENTS




































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2102 HGajst et al






























APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



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dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

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dα

dt= Cd


] (1)


E (x t) = E0 [1 minus α (x t)] (2)



σ (t) = ε0 (t)

(3)

where = int10



ε (x t) = ε0 (t)



dα

dt= Cdε



] (5)


D (α) = D1 exp (D2α) (6)



εcr = minus(

1 + ξ0

D1

) 1n

(7)


dα

dt= χε2


] (8)



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FR = σlowastAE


(minus D2n α

)σlowastMax

(9)


σlowastMaxsim=




dα

dt= χε2


] (11)





d


lowast (t) + d

dt(α0 (t)) (13)


δα (t) = e f (t)

int t

0

dα0 (t)





3 RESULTS





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Cd (σ ) = Cd0 + Cd1 exp (σσ0) (16)










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2098 HGajst et al





4 D ISCUSS ION



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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)





D2

n= minus E0

E0 minus Eeffln

(FR σMax

Eeffεcr

)(17)


Dow




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2100 HGajst et al





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5 CONCLUS IONS



ACKNOWLEDGEMENTS




































Dow




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2102 HGajst et al






























APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



Dow




ay 2020



dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

Dow




ay 2020





FR = σlowastAE


(minus D2n α

)σlowastMax

(9)


σlowastMaxsim=




dα

dt= χε2


] (11)





d


lowast (t) + d

dt(α0 (t)) (13)


δα (t) = e f (t)

int t

0

dα0 (t)





3 RESULTS





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2096 HGajst et al






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Cd (σ ) = Cd0 + Cd1 exp (σσ0) (16)










Dow




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4 D ISCUSS ION



Dow




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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)





D2

n= minus E0

E0 minus Eeffln

(FR σMax

Eeffεcr

)(17)


Dow




ay 2020

2100 HGajst et al





Dow




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5 CONCLUS IONS



ACKNOWLEDGEMENTS




































Dow




ay 2020

2102 HGajst et al






























APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



Dow




ay 2020



dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

Dow




ay 2020

2096 HGajst et al






Dow




ay 2020



Cd (σ ) = Cd0 + Cd1 exp (σσ0) (16)










Dow




ay 2020

2098 HGajst et al





4 D ISCUSS ION



Dow




ay 2020


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)





D2

n= minus E0

E0 minus Eeffln

(FR σMax

Eeffεcr

)(17)


Dow




ay 2020

2100 HGajst et al





Dow




ay 2020



5 CONCLUS IONS



ACKNOWLEDGEMENTS




































Dow




ay 2020

2102 HGajst et al






























APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



Dow




ay 2020



dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

Dow




ay 2020



Cd (σ ) = Cd0 + Cd1 exp (σσ0) (16)










Dow




ay 2020

2098 HGajst et al





4 D ISCUSS ION



Dow




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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)





D2

n= minus E0

E0 minus Eeffln

(FR σMax

Eeffεcr

)(17)


Dow




ay 2020

2100 HGajst et al





Dow




ay 2020



5 CONCLUS IONS



ACKNOWLEDGEMENTS




































Dow




ay 2020

2102 HGajst et al






























APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



Dow




ay 2020



dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

Dow




ay 2020

2098 HGajst et al





4 D ISCUSS ION



Dow




ay 2020


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)





D2

n= minus E0

E0 minus Eeffln

(FR σMax

Eeffεcr

)(17)


Dow




ay 2020

2100 HGajst et al





Dow




ay 2020



5 CONCLUS IONS



ACKNOWLEDGEMENTS




































Dow




ay 2020

2102 HGajst et al






























APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



Dow




ay 2020



dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

Dow




ay 2020


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)





D2

n= minus E0

E0 minus Eeffln

(FR σMax

Eeffεcr

)(17)


Dow




ay 2020

2100 HGajst et al





Dow




ay 2020



5 CONCLUS IONS



ACKNOWLEDGEMENTS




































Dow




ay 2020

2102 HGajst et al






























APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



Dow




ay 2020



dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

Dow




ay 2020

2100 HGajst et al





Dow




ay 2020



5 CONCLUS IONS



ACKNOWLEDGEMENTS




































Dow




ay 2020

2102 HGajst et al






























APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



Dow




ay 2020



dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

Dow




ay 2020



5 CONCLUS IONS



ACKNOWLEDGEMENTS




































Dow




ay 2020

2102 HGajst et al






























APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



Dow




ay 2020



dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

Dow




ay 2020

2102 HGajst et al






























APPENDIX


13 = minus partF

partα

dα

dtminus Pe

dφ

dt (A1)


dφ


partF

partα(A2)

dα


partF

partα (A3)



Dow




ay 2020



dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

Dow




ay 2020



dα

dt= Cd


](A4)






dα

dt= Cd

[D (α ϕ) εn+2 + ε2 (1 minus ξ0)

] (A5)

Dow




ay 2020

Relating strain localization and Kaiser effect to yield ...

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