Obtaining the size distribution of fault gouges with polydisperse bearings

Transcription

1 Obtaining the size distibution of fault gouges with polydispese beaings Pedo G. Lind, Reza M. Baam, 2 and Hans J. Hemann 2, 3 Institute fo Computational Physics, Univesität Stuttgat, Pfaffenwalding 27, D-7569 Stuttgat, Gemany 2 Computational Physics, IfB, HIF E2, ETH Hönggebeg, CH-893 Züich, Switzeland 3 Depatamento de Física, Univesidade Fedeal do Ceaá, Fotaleza, Bazil (Dated: ovembe 9, 27) We genealize the ecent study of andom space-filling beaings to a moe ealistic situation, whee the spacing offset vaies andomly duing the space-filling pocedue, and show that it epoduces well the size-distibutions obseved in ecent studies of eal fault gouges. In paticula, we show that the factal dimensions of andom polydispese beaings sweep pedominantly the low ange of values in the spectum of factal dimensions obseved along eal faults, which stengthen the evidence that polydispese beaings may explain the occuence of seismic gaps in natue. In addition, the influence of diffeent distibutions fo the offset is studied and we find that the unifom distibution is the best choice fo epoducing the size-distibution of fault gouges. PACS numbes: d, n, 6.43.Bn I. ITRODUCTIO In the ealy nineties the question of the possibility to tile an abitaily lage stip of space-filling olle beaings without fiction no slipping was addessed [], motivated by the study of eal systems such as seismic gaps. Seismic gaps ae egions along a fault zone whee eathquakes do not take place and theefoe they could be explained by sheaed plates on a space-filling beaing [2]. Fault zones ae typically self-simila and the mechanical oigin of the powe-law in the paticle size distibution was associated to the paticle s factue pobability which has been poposed to be contolled by the elative size of its neaest neighbos [3, 4]. Moe ecently, a geophysical model [5] explained the diffeent values of the factal dimension, anging fom d f = 2.6 to d f 3, by taking into account the fault gouge stain. While such model explains the dynamical oigin of diffeent powe-laws, thee is still the question if the space-filling beaing scenaio is able to epoduce such empiical esults in a simple and systematic way. By epoducing with space-filling beaings the same paticle size distibutions obseved in fault zones one can stengthen the hypothesis that the existence of seismic gaps in fault zones may be elated to the emegence of paticula geometical aangements of thei composing ocks, due to local fagmentation duing tectonic motion. Figue illustates two andom space-filling beaings in two and thee dimensions. Pioneeing studies with space-filling beaings wee done using deteministic pocedues in two [] and thee [6] dimensions and also using andom algoithms [7]. Howeve, up to now specific initial configuations and fixed paamete values wee addessed. In this pape we pesent a geneal algoithm to constuct ealistic space-filling beaings that allows to epoduce the ange of factal dimensions obseved in fault gouges [3 5, 8]. The model is paameteized by a unique paamete that contols the stength of fagmentation and takes into account ensemble aveages. Despite the wide feedom in the paametes and initial configuations, the systems pesents obust esults in what concens the factal dimension. In paticula, we will show that by vaying the ange of admissible values of the contol paamete one finds factal dimensions FIG. : (Colo Online) Illustation of a andom space-filling beaing in thee and in two dimensions. Discs and sphees of the same colo do not touch each othe. The andom space-filling beaing stats with a lage disc o sphee, maximizing polydispesity (see text). obseved in fault gouges. We stat in Section II by descibing in some detail the pocedue to geneate andom space-filling beaings, intoducing a paamete that accounts fo fagmentation at the local scale. In Sec. III and IV we descibe the esults fo two and thee dimensions espectively, with special emphasis on the size distibution and the factal dimension. Discussions and conclu-

2 2 sions ae given in Sec. V. II. THE RADOM SPACE-FILLIG OF PARTICLES In this Section we will stat by evisiting pevious pocedues [7] fo constucting andom space-filling packings and beaings and then intoduce the necessay ingedients to obtain a fully andom space-filling beaing. Random beaings in two and thee dimensions ae constucted in the following way. Fist, one stats by andomly distibuting a small numbe of discs o sphees within a given ange of sizes, without touching each othe. Second, one fills the empty spaces in the system by intoducing iteatively the biggest possible disc o sphee in the neighbohood of some empty egion. Thid, one esizes some disc o sphee in ode fo the packing to be bi-chomatic (beaing condition), i.e. only two colos ae needed to colo all discs in such a way that no discs of the same colo touch each othe. This guaantees the beaing condition: paticles ae able to ole on each othe without fiction o slipping. Figues 2a and 2b give illustative examples of such andom beaings in two dimensions. The filling pocedue is done by choosing andomly a void within the inte-disc fee space and then fitting the biggest disc in it, i.e. fit the disc that touches the thee neaest discs in the neighbohood, as illustated in Fig. 2a. Fo the thee dimensional case one consides sphees touching the fou neaest neighbos. The coloing pocedue is done by attibuting a pope colo to the intoduced disc. In the case that the thee neighboing discs have the same colo one attibutes the othe colo to the new disc. Othewise, one chooses only one of the neighbos to be in contact with the new disc, and the new disc shinks to a size with adius = α ( α ), whee is the adius befoe shinking and gets a diffeent colo as the disk it touches. Figue 2b illustates this coloing pocedue. Paamete α is ou contol paamete. Fo constant α = one obtains the paticula case whee beaing cannot be guaanted. In this case, fustated contacts emege when paticles ae foced to otate [7], which would eventually lead to the fagmentation of the discs into smalle ones. Recently a new method to implement ealistic gain factue in theedimensional simulations of ganula shea was poposed [9], based on beakable bonds between paticles within a medium. We keep the model simple, by using instead the eduction facto α that mimics the effect of fagmentation: by shinking a paticle oiginally with fustated contacts we mimic its fagmentation into smalle paticles that will fill the empty space left afte the fagmentation. The algoithm descibed above was peviously[7, ] used in two and thee dimensions by fixing a given initial configuation with discs having sizes within a given fixed ange and also by using a fixed eduction facto α duing the filling and coloing stages. The density of the packing was studied as a function of the numbe of existing sphees as well as the cumulative paticle size distibution. It was found that the cumulative distibution obeys a powe-law, namely () n(q)dq d f, whee d f is the factal dimension of the beaing [2]. ext, we intoduce the additional points to stengthen pevious findings and impove the algoithm descibed above. Fist, thee is the statistical significance of the esults, and thei sensitiveness to initial configuations, i.e. to the initial ange [ δ/2, + δ/2] of sizes. This initial configuation may influence the polydispesity of the system and consequently the attained distibution afte filling the entie system. As we will see, lage numbe of initial discs typically influence how the density inceases duing the space-filling pocedue. To take this point into account we enable the constuction pocedue to stat with an initial configuation having a single abitaily lage disc (o sphee) and pefom ensemble aveages on a significant numbe of initial configuations. Concening the single initial lage disc, one should notice that it does not suffice one single disc to intoduce a second one, because in two-dimensions each inseted disc needs to have at least thee neighbos (fou neighbous fo thee-dimensions). Howeve, as illustated in Fig. 2c, such stating disc can be intoduced into the system by peviously distibuting a few vey small seed-discs in the system and then following the algoithm descibed above. The numbe of such seeds is small, and theefoe they do not affect significantly the cumulative size distibution. Thei ole is that the aveage distance between them is eventually of the ode of the system size, en- (c) FIG. 2: (Colo Online) Sketch of the constuction of a andom spacefilling beaing. A new disc (blue) is andomly inseted in the system and is shifted and enlaged to the maximal accessible size (dashed cicle) without ovelapping neighboing discs. Then, it is educed by a facto < α, keeping a single contact point with one of the neighbos and assuming the opposite colo. (c) To stat the space-filling with a lage disc (bown) one needs to place peviously a few small seeds (geen) in the system (gey) and then poceed as in and [see text].

3 3 (c) α= α= α= α.4.2 d f m (d) e+5 [.5,.] [.5,] [,5]... FIG. 3: Density and size distibution in two-dimensional andom space-filling packings. Density of the packing as a function of the total numbe of discs fo diffeent values of α =, and., togethe with the distibution of the adius of the discs. In both cases, one stats fom a fixed initial configuation of = 4 discs with adius in the ange [.8R, R] with R = being the adius of the system (see Fig. b). The minimal adius of the initial set of discs is indicated as m.8 (see text). Fitting of the powe-law ange in, the distibution () = b d f yields the factal dimension d f as a function of α plotted in the inset. In (c) one plots the density as a function of fo the initial sets of = 4 discs in [.5R,.R], [.5R, R] and [R, 5R], i.e. anges with the same inteval width but centeed aound diffeent values, namely aound =.75,.75 and 25 espectively and fixing α =. In the inset of (c) the density () is plotted fo initial anges [ δ 2, + δ 2 ] having the same cente =.75 but diffeent widths δ =,, and. In (d) we plot distibution () of the sphees as a function of fo the same conditions as in (c). In all cases = 5 discs. abling the intoduction of a fist disc with the size of the ode of the system size. Of couse, depending on the numbe and distibution of the seeds, the fist initial discs may have also a size within the initial ange of sizes. In this way, one genealizes the pevious pocedue [7] and maximizes the admissible polydispesity. Second, we also intoduced a citeion to incease computational efficiency of ou algoithm. The neighbohood wee the neighboing discs (o sphees) ae seached fo must be chosen conveniently. We popose to choose a size that deceases with the incease of the density, since the dense the packing the smalle ae the empty spaces to put new discs. Theefoe, the adius n of the neighbohood of a given andom point intoduced in the system at iteation n, is updated as n = n n ( sys max ), whee sys and max ae the adii of the system and of the biggest disc o sphee in it, espectively. Thid, we also conside the contol paamete α to vay andomly within a tunable ange of values. In paticula, we ague that though a tentative value of constant α could be obtained by analyzing samples of gouges in eal situations, one expects that a cetain ange of admissible values fo α is the most ealistic assumption. Indeed, we show that the typical ange of factal dimensions obseved in eal fault gouges is in this way epoduced. III. THE TWO-DIMESIOAL CASE We stat this section by addessing the two-dimensional andom space-filling beaing and systematically eviewing the behavio of the packing fo diffeent, but fixed, values of α and study the effect of fixed initial size anges (no maximal admissible polydispesity). Figue 3 shows fo this case the density () and cumulative size distibution () of twodimensional andom space-filling packings. Figue 3a shows the density as a function of the numbe of discs fo α =. (packing) and also fo α = and (beaings) sepaately, stating fom = 4 discs with adius in the ange [ δ 2, + δ 2 ] having =. and

4 4.6 α in [.,.9] α in [,] α in [.3,.7] α in [,] α d f FIG. 4: Aveaging the density and the distibution size () ove initial configuations stating fom lage discs ( R/2) and with α vaying in a ange [.5 α/2,.5 + α/2] with α =,, and. The inset of shows that the factal dimension in () d f is almost independent of α yielding d f.54 which is, within the numeical eos of eal fault gouges (see text). δ =.6. As expected, the convegence as inceases is faste fo lage values of α. We conside one fixed initial configuation with initial discs, and theefoe the diffeent cuves coincide fo <. In Fig. 3b we plot the distibution of the adius of the discs, whee m = δ 2 is the minimal adius of the initial set of discs, and the deviation fom the powe law fo > m is due to the initial configuation. Below this value m, the size distibution obeys a powe-law = b d f, whee d f is the factal dimension of the packing, plotted in the inset as a function of α (symbols). As one sees fom the inset, the factal dimension typically takes values in the ange.2 < d f <.4, diffeently fom the values found in two-dimensional cuts of fault gouges (.6 ±.). Thee is a maximum of d f fo α =.5 that can be explained fom the definition of α in the algoithm descibed above. Fo an α <.5, to each new disc intoduced thee is a emaining fee space chaacteized by α >.5 such that α + α =, and similaly fo α >.5. Both Figs. 3a and 3b conside the same initial configuation. To study the influence of the initial configuations, we plot in Fig. 3c the density (), fixing α =, simila to pevious woks[], and using diffeent size anges fo the initial sets of = 4 discs, namely in [.5R,.R], [.5R, R] and [R, 5R], i.e. anges with the same width δ =.6 but centeed aound diffeent values, namely aound =.75,.75 and 25 espectively. Since diffeent initial configuations ae now used, the density is no longe the same below as in Fig. 3a. Futhe, one obseves that the density conveges to one fo inceasing the value of. In the inset the density () is plotted by fixing =.75 and stating with initial configuation having diffeent widths, namely δ =,, and. In these cases the density gives always simila dependencies on. Theefoe, the aveage size of the initial configuation is the impotant paamete to tune the density of the packing. Its width can be vaied without changing significantly the esults. In Fig. 3d we plot the accumulative size distibution () of the discs fo the same conditions as in Fig. 3c. The value of the exponent emains almost constant, d f.35. In othe wods, the factal dimension is not vey sensitive to the initial configuation, and the paamete on which the factal dimension depends moe stongly must be indeed α. Since α is also the paamete contolling the fagmentation of discs with fustated contacts (see above), we will now study it moe deeply. When α is able to vay andomly, the fagmentation of the lagest disc in the fee holes can be egaded as a andom pocess by its own. We next conside α to be each time andomly selected fom a fixed inteval [α α/2, α + α/2]. We will show that when enabling α to take diffeent values fo each paticle shinking, one obtains factal dimensions simila to the ones obseved in fault gouges[3 5]. To this end, we put eveything togethe, namely α vaying in the middle ange of admissible values, a lage initial disc and an ensemble aveage ove a significant numbe of initial configuations. The esults fo density and size distibution

5 5 d f β FIG. 5: The factal dimension as a function of the exponent β when the value of α is chosen accoding to a powe-law P(α) α β in a ange α [.,.9]. Although within the eo bas, the factal dimension is somewhat lowe compaed to Fig. 4b whee the distibution of chosen α values is unifom (see text). ae shown in Fig. 4, whee one consides thee initial seeds in the ange M = 2 m R/, with R the size of the system and α vaies andomly in the ange [.5 α/2,.5+ α/2]. Aveages ove a sample of initial configuations. Since initialization, filling and coloing pocedue ae now all andom, we call these systems fully andom space-filling beaings. Figue 4a shows the density as a function of the numbe of discs fo α =,, and. One sees an abupt tansition above = 3 (initial seeds), due to the intoduction of the fist lage disc. Fo all the fou cases the dependence of on the ange of α-values is simila, with the convegence towads = being slightly slowe fo naowe anges, because they hinde the occuence of lage discs. Figue 4b shows the size distibution () fo each of the fou anges. All the distibutions almost coincide, as shown in the inset whee the factal dimension d f taken fom () d f is almost constant (d f.54). This value is lage than the one obtained when α is kept constant (see Fig. 3). otice that the value of the factal dimension fo α =, though coesponding to the case of constant α =.5, is diffeent fom the one plotted in the inset of Fig. 3b, since the constucting pocedue of the beaing is slightly diffeent (see Sec. II). Since the above value is obtained fom a significantly lage sample of initial configuations and all the paametes α and position of the discs ae andomly selected, we will conside this value d f =.54 as the chaacteistic exponent of the size distibution fo fully andom two-dimensional space filling beaings. The aveage chaacteistic value d f =.54 obtained lies in the ange of values measued of the factal dimension measued in eal fault gouges (D =.6 ±.) [3, 4], as indicated with a dashed line and shadow egion in the inset of Fig. 4b. Till now, the value of α was consideed to vay unifomly within a cetain ange of values. If the pobability distibution fo choosing α values is Gaussian simila esults ae obtained, wee the standad deviation σ of the distibution plays a simila ole as the width α used in Fig. 4. Howeve, if we take values within a ange, say α [.,.9], and chose them accoding to a powe-law distibution P(α) α β the factal dimension changes significantly, as shown in Fig. 5. Even fo small values of the exponent β, e.g. β =.5, the factal dimension deceases when compaed to the value obtained fo the unifom distibution (β = ) and emains appoximately constant at d f.43. otice that β = in Fig. 5 coesponds to α = in Fig. 4b. If the powe-law distibution selects values in a ange with a diffeent width, a simila decease of the factal dimension is obseved when compaing with the unifom distibution case. Theefoe, one can conclude that a easonable choice fo constucting space-filling beaings with factal dimension simila to the one obseved in fault gouges is by taking a andom value of α unifomly distibuted in a cetain ange aound.5. IV. THE THREE-DIMESIOAL CASE As descibed above in Sec. II, a thee-dimensional vesion of fully andom space-filling beaings is obtained in a simila way as fo discs with the single diffeence that the intoduction of new sphees takes into account fou neaest neighbos. In this Section we addess the case of thee-dimensional spacefilling beaings as a moe ealistic appoach to fault gouges, and study how well the two-dimensional model appoximates thee-dimensional systems of sphees. Recently [9], it was found that gain factue simulations d f α FIG. 6: The factal dimension as a function of α fo theedimensional andom space-filling beaings when α is kept constant. A simila pocedue as the one illustated in Fig. 2 is used, with new sphees being intoduced touching the fou neaest neighbos (see text). The dashed line indicates one typical value d f 2.58 found in some eal fault gouges [9].

6 6 α= α= α= α= d f 2.4 α FIG. 7: Thee-dimensional space-filling beaings fo vaying α [.5 α/2,.5 + α/2]. The density as a function of the numbe of sphees fo α =,, and and the coesponding size distibution () with the factal dimension d f in inset. In all cases = 4. poduce a comminuted ganula mateial simila to the one obseved in eal fault gouges. Fom those simulations, it followed that comminution ate and suvival of lage gains is sensitive to applied nomal stess, with a factal dimension of the esultant gain size distibutions in the ange d f [2.3 ±.3, 2.9 ±.5], that agees with the obsevations of thee-dimensional samples of eal gouges whee typically d f In thee dimensional space-filling beaings with a constant value of α the factal dimension lies above the obseved values in fault gouges. In Fig. 6 we plot typical values of d f as a function of α. The factal dimension of such beaings is typically lage than d f 2.58 (dashed line) with values within the ange d f [2.6 ±., 2.74 ±.5] and, similaly to the two-dimensional case, the maximum of d f is eached fo α.5. As summaized above in the intoduction, it was ecently found [5] that factal dimensions 2.6 ae obseved fo lowstain gouges. In egions subject to lage shea stain the factal dimension is significantly lage, 3. Theefoe, the paticle size o mass dimensions wee poposed as a way to distinguish between egions with diffeent stain stengths [5]. Fom Fig. 6, one sees that a simila ange of values fo the factal dimension is also found fo space-filling beaings. Futhemoe, the explanation elating the factal dimension of fault zones and thei stain stength assumes that fagmentation is contolled by neaest neighboing paticle contact. α d f α= α= α= α= α=... FIG. 8: The size distibution of two-dimensional cuts of the theedimensional space-filling beaings addessed in Fig. 7 and the coesponding factal dimension. and that a paticle is most likely to split into smalle paticles with a paticle of simila size, yielding a lage factal dimension 3. In the case of ou constuction pocedue fo space-filling beaings this would coespond to the case of α.5. Indeed, fom Fig. 6 one obseves that the maximum

7 7 of the factal dimension is eached fo such α values yielding d f = 2.74 ±.5. When vaying α andomly in a ange aound.5 and study the dependence of the space-filling beaing on the width α of the ange [.5 α/2,.5+ α/2]. In Fig. 7a one sees that the density inceases faste fo lage α, similaly to what was shown in Fig. 4a. As fo the factal dimension, Fig. 7b shows that it deceases slightly when compaed with the case of constant α ( α = ). Theefoe, inceasing the width α of the ange of admissible values fo α one is able to educe the factal dimension of the beaing. Similaly to the situation of measues taken in fault gouges, the two-dimensional coss section of such thee-dimensional space-filling beaings should have a factal dimension within the ange of the obseved empiical values in eal fault gouges (d f =.6 ±. [3, 4]). By aveaging seveal diffeent two dimensional coss sections of the 3D beaings we plot in Fig. 8a the size distibution of a typical two-dimensional coss section fo the diffeent values of α. In Fig. 8b we obseve that only fo vey wide anges of α values it is possible to obtain a factal dimension simila to the one obseved on fault zones. V. DISCUSSIO AD COCLUSIOS In this wok we studied the size-distibution of andom space-filling beaings with lage polydispesity, showing that it epoduces well the size-distibution found in fault gouges. Focusing on the dependence of the beaings factal dimension on the spacing offset, we have shown that the factal dimensions of such beaings sweep the low ange of values obseved in eal faults. Since ecently it has been epoted that the factal dimension vaies in space along fault gouges[5], ou findings enables us to conjectue that the occuence of seismic gaps, whee eathquakes ae absent and theefoe behave similaly to olle beaings, may occu in egions whee the factal dimension lies in the low ange of admissible values, namely d f [2.5, 2.75]. To compute an accuate value fo the exponent chaacteizing andom beaings, we intoduced a geneal algoithm that allows α to vay andomly in a wide ange of admissible values, typically < α < and stat the space-filling pocedue fom one unique lage disc (o sphee), maximizing the ange of admissible sizes in the beaing. With such model we wee able to show that beaings have a factal dimension with values within the ange of values in eal fault. Since it is known [5] that along a specific fault gouge the factal dimension vaies typically between 2 and 3, ou esults suppot the hypothesis that seismic gaps, occuing only in cetain paticula locations of the fault, could be explained by this simple geometical model. Futhe, we also intepet the contol paamete α fo the beaing popety as a measue of the fagmentation stength, and intoduce simple citeia to impove the computational efficiency of pevious space-filling packing algoithms. To impove futhe ou findings we should also take the effect of gavity into account. Moeove, concening the spacefilling beaings by themselves, othe questions aise, namely thei contact netwok coelations, which should help to undestand the ange of obseved factal dimensions. These and othe points will be addessed elsewhee. Acknowledgments The authos thank Bibhu Biswal fo useful discussions. This wok was suppoted by Deutsche Foschungsgemeinschaft, unde the poject LI 599/-. [] H.J. Hemann, G. Mantica and D. Bessis, Phys. Rev. Lett. 65, (99). [2] J.A. Astøm, H.J. Hemann and J. Timonen, Phys. Rev. Lett. 84, (2). [3] C. Sammis, G. King and R. Biegel, Pageoph. 25, (987). [4] C. Sammis and R. Biegel, Pageoph. 3, (989). [5] C.G. Sammis and G.C.P. King, Geophys. Res. Lett. 34, L432 (27). [6] R. Mahmoodi Baam, H.J. Hemann and. Rivie, Phys. Rev. Lett. 92, 443 (24). [7] R. Mahmoodi Baam and H. Hemann, Phys. Rev. Lett. 95, (25). [8] S. Roux, A. Hansen, H.J. Hemann and J.-P. Vilotte, Geophys. Res. Lett. 2, (993). [9] S. Abe and K. Mai, Geophys. Res. Lett. 32, L535 (25). [] R. Mahmoodi Baam and H.J. Hemann, pepint 24. [] J.H. Steele, Metall. Tans. A 7, 325 (976). [2] S.S. Manna and H.J. Hemann, J. Phys. A 24, L48-L49 (99).