Traveling wave solutions in the Burridge-Knopoff model C. B. Muratov Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012 (February 9, 2008) The slider-block Burridge-Knopoff model with the Coulomb friction law is studied as an excitable medium. It is shown that in the continuum limit the system admits solutions in the form of the self-sustained shock waves traveling with constant speed which depends only on the amount of the accumulated stress in front of the wave. For a wide class of initial conditions the behavior of the system is determined by these shock waves and the dynamics of the system can be expressed in termsoftheirmotion. Thesolutionsintheformoftheperiodicwavetrainsandsourcesofcounter- propagating waves are analyzed. It is argued that depending on the initial conditions the system 9 will either tend to synchronizeor exhibit chaotic spatiotemporal behavior. 9 9 PACS Number(s): 83.50.Tq, 91.30.Mv, 83.50.By, 03.40.Kf 1 n a J I. INTRODUCTION porting steadily propagating solitary waves in the form 4 of shocks [7–10]. These systems are also of special in- 1 Propagating self-sustained waves (autowaves) and terest because they are used for modeling the dynamics more complex spatiotemporalpatterns arecharacteristic of earthquakes[7–17]. It is clear that systems exhibiting 1 v of excitable media of different nature. A typical exam- stick-slip motion have both necessary ingredients of ex- 3 ple of such a phenomenon is burning of black powder citable systems. The threshold behavior here is due to 0 in a safety fuse. When the fuse is ignited at one end, static friction, which prevents any motion in the system 0 the exothermic reaction releases the heat which is then until some critical amount of stress is accumulated. The 1 spread out by heat diffusion. Thus, the neighboring re- couplingoftheelementsofthe systematdifferentpoints 0 gionsofthe fuse igniteleadingto self-sustainedpropaga- in space is due to the non-locality of elastic stress. 9 tion of the combustion front. The phenomenon of non- Singularperturbationtechniques provedto be veryef- 9 / attenuated propagation of waves is in fact common for fectiveintreatingproblemsoftravelingwavepropagation ol a variety of physical, chemical, and biological systems in reaction-diffusionsystems [1–6,18,19]. These methods s [1–6]. Traveling waves are experimentally observed in usestrongseparationoftimescalesintheproblemtode- - semiconductorandgasplasma,semiconductorandsuper- compose the dynamics of the system into fast and slow t at conductorstructures,combustionsystems,activeoptical motion. Clearly,thissituationisalsorealizedinthemod- p media,magneticmediaunderillumination,autocatalytic elsofstick-slipmotionwhere(especiallyinthecontextof : chemical reactions, nerve and heart tissue (see [1–6] and earthquakes) there is a strong separation of time scales v i references therein). between fast slipping events and slow accumulation of X Inorderforself-sustainedwavestobefeasible,thesys- stress. It is therefore advantageous to try to apply these r tem must possess two basic ingredients. First, the sys- techniques to the problem of stick-slip motion. a tem must be excitable, that is, there has to be a thresh- In this paper we present a study of the Burridge- old below which the perturbation of the steady homo- Knopoff slider-block model [11] with the Coulomb fric- geneous state of the system decays, while perturbations tion law. We will show that for sufficiently slowly spa- of larger amplitude grow. In the example above, a suf- tially varying displacement variable the dynamics of the ficient amount of heat is needed to ignite black powder. system is dominated by self-sustained traveling shock Second, there has to be coupling between the regions waves. We will study the properties of these waves and of the system at different points in space. In the case reformulatethe dynamics of the systemin terms oftheir of the safety fuse such a coupling is provided by heat motion. diffusion leading to spread of the temperature and ig- Our paper is organized as follows. In Sec. II we intro- nition of powder in front of the combustion zone. Thus, ducethegoverningequationsforthemodelwestudyand theprototypesystemsexhibitingself-sustainedwavesare discuss the features of the friction law used, in Sec. III reaction-diffusion systems [1–6]. we construct the solutions in the form of self-sustained Recently, it was pointed out that an entirely differ- traveling shock waves, in Sec. IV we reformulate the ent class of systems may be considered as excitable [7]. dynamics of the system in terms of the motion of these Theseareelasticmediawithfrictionexhibitingstick-slip shock wavesand study generalproperties of the reduced motion. Both experimental observations and numerical problem, in Sec. V we analyze two different types of simulations show that such systems are capable of sup- solutions, and in Sec. VI we draw conclusions. 1 II. MODEL f The Burridge-Knopoff model consists of a one- ∼αv dimensional array of blocks of mass m resting on a fric- f tionalsurface[11]. Theblocksareconnectedtogetherby r springs with spring constant k and pulled by a loader c plate moving with constant speed V via another set of f springs having spring constants k (see Fig. 1). Let us s p V a a v k k k p p p FIG.2. Friction force as a function of thevelocity k k c c that this model was recently studied numerically in Ref. m m m [17] (although for the parameters different from those (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) considered in this paper). (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) Wewouldliketoemphasizethatincontrasttovelocity- (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) FIG.1. The Burridge-Knopoff model weakening friction law, for which the kinematic friction force is a continuous function of the block velocity at measurethedisplacementXi ofthe i-thblockrelativeto v = 0, in our friction law the force f has a discontinu- the point ofattachmentof the i-thloaderspring. In this ity at v = 0, reflecting the fact that the coefficient of case the total force Fi acting on the i-th block is given static friction is typically larger than that of kinematic by friction. As a result of this difference, the accumulated stressacceleratestheblockattheonsetofmotion,releas- Fi =kc(Xi+1+Xi−1 2Xi) kpXi fi, (1) ing potential energy. As we will see below, this provides − − − the sustaining force for the traveling waves studied in wheref istheforceoffriction. Thedynamicsofthesys- i the following section. Also, as we will show below, this tem is completely determined by the equation of motion ensures the existence of the proper continuum limit as mX¨ = F , provided that the friction law is specified. i i a 0. Note that the friction force f is the only nonlinearity → i Let us now formulate the equations of motion for the in the equation of motion. Clearly, the dynamics of the blocks in the case of the friction law introduced above. system will significantly depend on the particular choice RecallthatthedisplacementsX aremeasuredrelativeto i of the friction law. Recently, a lot of results were pre- theloaderplate,sotheyaredefinedinthereferenceframe sentedonthedynamicsoftheBurridge-Knopoffmodelin movingrelativetothesurfacewithvelocityV. Therefore, the case of velocity-weakening friction law [9,10,12–16]. when the blocks are at rest, their equation of motion A characteristic feature of the Burridge-Knopoff model becomes with this form of friction is its highly chaotic dynamics that occurs on all length scales down to the smallest X˙ = V. (2) i − length scale a and therefore, the absence of the proper continuum limit [12]. When the block slides,its equationofmotionchangesto In contrast to most previous studies, here we adopt the Coulomb friction law, which is applicable to clean mXï =kc(Xi+1+Xi−1−2Xi)−kpXi−fs−α(X˙i+V). dry surfaces (see, for example, [20]). Namely, we will (3) characterize the friction force by the value of the static friction fr and the kinematic friction fs < fr, where fr The transition to sliding (slip) occurs when the force Fi andfsarepositiveconstants. Thus,theblockwillremain reaches the value of fr: atrestifF 0 [see Eq. (1), in order for slip Finally,letusdiscusstheapplicabilityoftheBurridge- i to occur,the force F mustexceed the value f of sliding Knopoff model in the context of real physical systems i s friction],while the blockswillalwaysslide foru >1[see exhibiting stick-slip motion. In a realsystem one should i Eq. (4)]. replace a one-dimensional array of masses between the Ifk k ,onecannaturallygotothecontinuumlimit loader plate and the frictional surface by an elastic c p bTyhimsaisk≫ianggoaosdubasptpitruotxiiomnaXtioi+n1w+hXeni−t1h−e l2eXngit→h sac2a∂l∂e2xX2of. msioendiuofmthoef cBerutrariindgteh-iKcknnoepsosffh.mAodsetlrawigohutlfdortwhaerrdefoerxetebne- variation of X is of order 1 in the new variables. Using a two-dimensional array of masses connected by springs i the variables in Eq. (5), we rewrite Eqs. (2) and (3), whose one edge is rigidly attached to the loader plate respectively, as follows: and the other slides on the frictional surface. Naturally, thethicknessofsuchamediumshouldgreatlyexceedthe u =v, (6) microscopiclengthscalea. The stick-slipmotionin such t a medium is due to surface waves [26]. The dispersion u =u u 2γ(u v), (7) tt xx− − t− of these waves is given by ω2 = k2 +(π/2h)2(1+2n)2, where we introduced dimensionless constants where n is an integer and the transverse speed of sound was taken to be 1. The main difficulty here is that in- α V kpm stead of a single displacement variable u one has to deal γ = , v = , (8) 2 kpm frp−fs wficitahtioanlatrhgaetnisumprbeesrenotfemdobdyetshweiBthurdriiffdegree-nKtnno.pAoffsmimopdlei-l and dropped the pprimes for simplicity of notation. Note consists of lumping up these modes into a single mode that in the continuum limit in addition to tracking the withaneffectivestiffnesskp [Eq. (3)]. Itisclearthatthe motion of individual blocks, one also has to follow the dominant modes will be those with small n, so we must motionoftheslippoints. Similarly,oneshouldkeeptrack have kp (a/h)2kc kc. Thus, in a physically rele- ∼ ≪ ofthepositionsofthepointsatwhichtheblockscometo vant situation one should consider the continuum limit rest. The latter are determined by the condition in Eq. ∆x 0 of the Burridge-Knopoffmodel. In short, this is → (6) during sliding. merely a simplification of the mathematical handling of Determining the position of the slip points in the con- the elastic dynamics. A much more important assump- tinuum limit turns out to be a rather complicated prob- tion is that the friction response to the motion of the lem, since the positionofthe slipwillstill dependonthe mediumisinstantaneousandhaszerocorrelationlength. local dynamics of the blocks. In the discretized form, It would be incorrect to think of the “size” of the block the slip condition [Eq. (4)] in the variables of Eq. (5) a as an atomic distance. Rather, the value of the pa- becomes rameter a should have a magnitude of the correlation (memory) length of the cooperative effects responsible ui+1+ui−1−2ui =u 1, (9) for the surface friction (see, for example, [21] and refer- (∆x)2 i− ences therein). Then the response of the system can be considered to be instantaneous if the characteristic ve- where ∆x = kp/kc. We will get back to this problem locity of the blocks is greater than a/τstick, where τstick in the following section where we will show that in the is the characteristic sticking time. p continuum limit the dynamics of the slip becomes inde- pendent of ∆x. Equations(6)–(9)arethe constitutiveequationsthat III. TRAVELING WAVES will be studied in the rest of the paper. As can be seen from Eq. (5), we chose time and length scales so that LetusnowdemonstratethatEqs. (6)–(9)admitsolu- the speed of sound in the system is equal to 1. The time tions in the form of traveling shock waves with constant scaleisdeterminedbytheperiodofoscillationsofaniso- speed c in the limit of vanishingly small v. But before lated block due to the loader spring. Note that in the we do that, let us see what kind of dynamics one would continuum limit the characteristic speed of a block dur- observe if there are no slip events and the initial distri- ingslidingismuchsmallerthanthespeedofsound. This bution of u is taken to be a sufficiently slowly varying canbeseenfromEq. (5)ifoneassumesu 1,measures t ∼ distribution u0(x) (the latter ensures that no slip events X in the units of Eq. (5), and uses a natural assump- i will occur at short times). In this case the dynamics is tion that f f ak . The behavior of the system is r − s ≪ p trivial: we will have u(x,t) = u0(x)+vt, as long as no determined only by two parameters: the dimensionless slipeventsoccur[seeEq. (6)]. Fromthisonecanseethat dissipationparameterγ whichmeasuresthe effectofvis- the characteristic time scale of accumulation of stress is cousfriction,andthe dimensionlessrateofaccumulation v−1 1, when v is small. In other words, for v 1 on ofstressv. Inthefollowingwewillconsidervtobesmall, ≫ ≪ 3 thetimescaleoforder1onewillnotseeanymotionwith 3 2.8 thedistributionofufixedtou (x)+C,whereC isacon- 0 2.6 stant. Inparticular,ifonestartswiththe uniforminitial 2.4 conditions, one will have u=u , where u is some con- + + 2.2 stant less than 1. The other possible situation when the c 2 dynamics of the system becomes trivial is steady creep, 1.8 when all the blocks move together with the loader plate, 1.6 so we simply have u=2γv 0 for v 1 [see Eq. (7)]. 1.4 ≃ ≪ Consider the system with u = u and v = 0. As was + 1.2 noted in the previous section, when u+ < 0 slip events 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 cannot occur, so if one slightly moves a particular block, u + it will quickly move to a new equilibrium position and FIG. 3. The dependence c(u+). Results of the numerical no significant changes will occur. A different situation is solution ofEqs. (A3)–(A7). Dashedlineshowstheresultof realized for 0 < u < 1 when the system becomes ex- + approximation by Eq. (11). citable. Then, if a single block is slightly moved, it will accelerate,releasingthe potentialenergyaccumulatedin u . As a result of this acceleration, the force acting Toactuallycalculatethespeedcofthefront,oneneeds + on the adjacent blocks will increase resulting in slipping to solve the system of coupled equations of motion for of the adjacent block that was at rest. This avalanche- the blocks behind the s-th block. This problem in the like process will go on, leading to the formation of the limit ∆x 0 is considered in Appendix A. The analy- → shock wave that releases the accumulated stress, so that sis of this problem shows that the speed c of the wave u changes to a new lower value of u−. The situation is uniquely determined by the value of u = u+ in front hereresemblesagreatdealcombustionofsafetyfusedis- of the wave. Note that similar situation takes place in cussedintheIntroduction. Thus,for0 speed of individual blocks in the wave. 0. Because of the reflection symmetry, for each solution As can be seen from Fig. (3), the speed of the travel- traveling with speed c there is also a solution traveling ing wave diverges as u+ approaches 1. One can get an with speed c. analytical handle on the dependence c(u+) by expand- − Itisclearthatthespeedcofthetravelingwaveshould ing it in the inverse powers of c (see Appendix A). As a be determined by the motionof the slip point in frontof result, we get the following interpolation formula which the wave. Let the slip event occur for block s at t = ts. implicitly determines c as a function of u+ Then, for i > s we have ui = u+, so the slip condition 2+c2+4c4 −1 from Eq. (9) becomes u = 1+ . (11) + 8c5√c2 1 us−1(ts)=u+−(1−u+)(∆x)2. (10) Thisequationgive(cid:18)sthedepende−nce(cid:19)c(u+)withafewper cent accuracy. Since atthe onsetofthe slipu =u anddu /dt=0,at s + s timetwewillhaveu =u + (t t )2 ,withu 0,willalsotravelwithspeedcbehindthewave. th block. According to Eq. (10), this will happen when Behind the slip the blocks slide according to Eq. (7), so ∆t = t t = (∆x). Therefore, on the time scale s+1 s − O for the wave with speed c we will have of order 1 this will correspond to the motion of the slip point with the speed c=∆x/∆t= (1). (c2 1)u 2cγu +u=0. (12) O zz z − − 4 u u a) 1 u b) 1 u + + c c -w 0 z 0 z u - -1 -1 FIG.4. Two typesof traveling wavesolutions: (a) case u+ uc. The slip condition gives the following initial conditions for this equation: πγc u− = u+exp . (16) u(0)=u , u (0)=0. (13) − − c2(1 γ2) 1! + z − − p TheanalysisofEq. (12)(seeAppendixB)showsthatthe From this equation one can see that u− < 0, so right structureofthesolutionchangesqualitativelydepending behind the wave the system is in the state in which no on whether the value of u+ is smaller or larger than the waves can be further excited. Also, note that when u+ critical value uc > 0, where the value of uc is implicitly approaches the value of uc, the value of u− rapidly ap- determined by proaches zero. In the context of earthquakes a shock wave should be c2(u ) 1 associatedwith an individual earthquake. The total dis- c γ = − . (14) p c(uc) placement A=u+−u− which occurs upon passing of a wave and which determines the magnitude of the earth- If we have u+ < uc, the distribution of u will asymp- quake will therefore be totically approach u− = 0 at z = behind the wave −∞ [Fig. 4(a)]. πγc Inotherwords,theblocksbehindthewavenevercome A=u+ 1+exp . (17) " − c2(1 γ2) 1!# to rest. Therefore, upon passing of the wave the system − − shoulddevelopasteadycreepforsmallbutfinitev. Note p that this will always happen when γ > 1 since in any case u < 1, so for these values of γ the propagation of + IV. FREE BOUNDARY PROBLEM multiple shock waves becomes impossible (see Sec. V). Adifferentsituationtakesplaceforu >u . Thenthe + c Sofarwewerestudyingtravelingwavesolutionsinthe solution has the form shown in Fig. 4(b), and the width limit v = 0 and u = const in front of the wave. These w of the travelingwave becomes finite (see Appendix B) solutions are clearly the fast motions in the systemsince π(c2 1) they occur on the time scales of order 1. On the other w = − . (15) hand, although the accumulation of stress occurs slowly c2(1 γ2) 1 − − for v 1, it can lead to significant changes in u at long Thus,inthiscasethepwavepropagatesasa“self-healing” times≪of order v−1 1, so the latter can be associated ≫ pulse (compare with [28]). Note that for γ > 0 we have with the slow motions in the system. Therefore, since c > c(u ) > 1, so the value of w is bounded from below the fast motions result in the large-scale changes in u, c andthewidthofthewaveisalwaysgreateroroforder1. the slip-stick events can be considered as singular per- This justifies the use of the continuum limit for k k turbations to the slow motions. c p ≫ and γ 1. Furthermore, when u becomes close to 1, Similar situation is realized in other excitable media + ∼ the propagation speed c and therefore the width of the where the role of singular perturbation is played by the wavegrow,sointhiscasethecontinuumlimitisjustified sharp fronts [1–6]. A powerful tool in describing the dy- even for k k . On the other hand, the width w goes namics of such systems is reduction to free boundary c p ∼ to zero when u becomes small when γ 1 [see Eq. problem (see, for example, [18]). In this approach one + ≪ (15)], so for sufficiently small u and γ the continuum obtains the relationship between the speed of the sharp + limit will become invalid for the description of traveling frontandslowvariables. Usingthisidea,letusformulate waves. the dynamics of our system as a free boundary problem. In the case u+ > uc the value of u = u− behind the Let us assume that the distribution of u varies on the wave will be (see Appendix B) length scale much greater than 1. Then, introducing an 5 u adiabatic approximation, we can assume that the speed c of the traveling wave is given by the dependence c(u ) + (Fig. 3) in which now, instead of u one should use an + instantaneous value of u right in front of the wave. The latter will play the role of the slow variable. Thus, the motion of the shocks can be described by the variables x x (t) which give the positions of the i-th shock, and the i variables s = 1 which give the direction of their mo- i ± tion. On large length scales the shock will contribute a dis- continuity to the distribution of u in the limit v 0. It → canbeincludedinEq. (6)describingslowmotionsinthe FIG.5. Periodic wave train. form of a δ-function. Therefore, in the presence of the shocks Eq. (6) can be rewritten as The condition of the absence of creep should in fact be satisfiedfor a wide classof initial conditions. In particu- u =v c A(u)δ(x x +0s ), (18) t − i − i i lar, it is easy to see that a wave with u+ >uc will never Xi be able to reach the points where u u , so that the shock waves are always the considerations. Thus, there exists a family of nonlinear i c waves of switching between the blocks with u = u at periodictravelingwavesolutionswhosespeedandperiod + (cid:0) (cid:1) rest in front of the wave to u = u− at rest behind the dependstronglyonamplitude. Similarly,theperiodT of wave. In other words, we do not consider the possibility thissolutionintime depends onthe amplitude simplyas of the onset of creep behind the wave. The analysis of T = A/v, so the frequency of the shocks at a particular creep motion is beyond the scope of the present paper. point as the wave train passes is ω A−1. The latter in ∼ 6 u 0.2 0.1 2 v ’a/ x 0 -0.1 0 0.5 1 1.5 2 FIG.6. A source of counter-propagating waves. a/v2 FIG.7. Dependencea′(a) from Eq. (26) at γ =0.3. fact represents a simple form of the Gutenberg-Richter scaling law with scaling exponent b = 1 [29]. In other tice that Eq. (24) in general shows how to treat the sin- words, this exponent would be observed in the system gularity in the free boundary problem (Sec. IV) associ- under consideration if its dynamics were dominated by atedwiththe creationofa apair ofcounter-propagating periodic wave trains. It is interesting to note that the shocks. value of b actually observedfrom the earthquakes gener- The solution for xi(t) in Eq. (24) then allows to cal- ′ ally lies in the range 0.8 uc, so that no creep develops behind the traveling fr+ont. u′(x,t)≃u−(1)−a′(x−x0)2+vt, (25) Therefore, according to Eqs. (17), and (21), traveling ′ where a is given by (see Appendix C) wave trains exist only when L>L , where min Lmin =v−1c(uc)uc. (22) a′ = aκ+ 2av(1+κ) , (26) − v+√v2+8a Let us now consider another kind of spatiotemporal patternsthatistypicalofexcitablesystems—thesource u−(1)= exp( πγ/ 1 γ2) [see Eq. (16)], and − − − ofcounter-propagatingwaves[1–6]. Theexistenceofsuch p asolutionisassociatedwiththefactthatifthereisanin- πγ πγ κ= 1+ exp . (27) hmoummogoefnueiosulsodcaistterdibauttxion=oxfu,athternesattssuocmhethpaotinthteinmtaimxie- (cid:18) (1−γ2)3/2(cid:19) − 1−γ2! 0 the value of u(x ) may reach 1, so that the blocks will p 0 From Eq. (27) one can see that the value of κ lies in the become unstable creating a pair of counter-propagating ′ range 0 < κ < 1. The plot of a as a function of a at a shocks(Fig. 6). After theseshocksmovedawayfromx , 0 particular value of γ is presented in Fig. 7. the distribution of u can once again have a maximum at From Eq. (26) one can see that there is a maximum x = x0, so after some time the value of u will increase of u′ at x = 0 (a′ > 0) when the shocks have passed, if until it reaches 1 and the cycle repeats. a 0, after the with,butwithadifferentvalueofa. Therefore,Eq. (26) slipoccurred,thusdeterminingthevelocityoftheshocks. defines an iterative map for the value of a correspond- The analysis of the free boundary problem in this case ing to the solutions for u at times nT, where n is the (Appendix C) then shows that right after the slip the number of iterations. Physically, a source with period T positions of the shocks will be given by will form at x = 0 creating traveling waves propagating away from it. Note, however, that even though the pe- v+√v2+8a riod of the source is a constant that does not depend on xi x0 √bt, b= , (24) the initial conditions, the solution represents an aperi- ≃ ± 2a odicprocesssinceaftereachcyclethevalueofachanges. where the plus and the minus correspond to the shocks The latter is represented in Fig. 7. Observe that we al- propagatingtotherightandtotheleft,respectively. No- ways have a′ < a. From Fig. 7 one can also see that 7 after many periods the value of a will tend to zero. The want to calculate the dependence of the wave velocity analysis of Eq. (26) (see also Fig. 7) shows that zero is on the amount of the accumulatedstress u by formally + ′ the only fixed point of the map a a for any γ. For combining Eq. (12)at z =0 with the continuum version → smallenoughvaluesofaEq. (26)canbeapproximatedas of Eq. (9) at z = 0. This would give an incorrect result a′ a[1 2a(1+κ)/v2]. Itistheneasytoshowthatafter c = 1/√1 u . This means that although the contin- + n ≃1 it−erationswe willhavea v2/[2n(1+κ)],where uum appro−ximation may be valid for finding the wave n ≫ ≃ a is the result of n iterations of the map. Note that profile behind the slip point, one still needs to solve the n because of the fact that a becomes smaller and smaller discrete problem at the tip of the wave in order to find witheachcycle,thedistributionofubecomesflatterand its velocity. flatter [see Eq. (23)], what means that after a long time In the limit of vanishing loader plate velocity the only the regions in the neighborhood of the source will tend parameter in the model is the dimensionless coefficient to synchronize. The characteristic size R of a region in of viscous friction γ. All our analysis was performed for which the synchronization takes place can be estimated arbitrary values of γ, so it is also applicable to the case as R a−1/2, so we will have R t1/2. Also note that γ =0,i.e.,whentheviscousfrictionisabsent. Thiscase, n forth∼esamereasoniftherearetw∼osourceswithunequal however, has one special feature. The thing is that for values of a < amax some distance R apart, after time γ = 0 we have uc =0 [see Eq. (14)], so as u approaches t R2theonewiththesmallervalueofawilloverwhelm zero the speed of the wave approaches 1 and the width th∼e one with the larger value of a. It is then natural to w goes to zero [see Eq. (15)]. Therefore, for a fixed expectthatiftheinitialdistributionofuhasallmaxima value of ∆x 1 the continuum limit will no longer be ≪ with a < a , the system will eventually synchronize justifiedforthedescriptionofthewaveswhenubecomes max into uniform oscillations in which all the blocks slide at sufficientlyclosetozero. Also,whenγ =0theamplitude once. A = 2u+ of these waves [see Eq. (17)] will be able to Different situation is realized when a > a . In this become arbitrarily small. max casethedistributionofuformingaftertheshocksmoved According to the results of our analysis, the solution away from the origin will be a minimum rather than a in the form of the traveling shock wave, as well as a pe- maximum of u, so the next creation of a pair of shocks riodic wavetrainofa givenperiod, is unique in the limit will not occur at x = x , but rather at different points of vanishingly small loader plate velocity, and its speed 0 in space. Therefore, it is natural to expect that for suf- is uniquely determined by the value of u in the front of ficiently random initial conditions one would see shocks the wave. This is in contrast with the results of Langer created and annihilated at random points in space, cre- and Tang [15] and Myers and Langer [16] for the models ating a kind of spatiotemporal chaos. with velocity-weakening friction where they find a con- tinuousfamily ofsuchsolutionsandthereforea selection problem. VI. CONCLUSION Notethatthesystemswithvelocity-weakeningfriction transform to the model considered by us in the limit of small V , where V is the characteristic velocity of vari- Let us now summarize the results of our analysis and f f ation of the friction force. Indeed, if the characteristic discusstherelationshipofourresultswiththoseobtained speed of a block during the slip exceeds V , the friction for other models of stick-slip motion. In the present f force will drop very rapidly, acting in the same way as paper we analytically investigated the Burridge-Knopoff in our model. To obtain the criterion of smallness of slider-block model with Coulomb friction law, which is V , we note that the effect of the velocity dependence of different from the one used in most previous studies f the frictionis mostpronouncedinthe slipregion,where, [9,10,12–16]. This difference is expressedinthe fact that according to Eq. (10), u ∆x= k /k , where we as- ourfrictionisadiscontinuousfunctionwhenthevelocity t p c ∼ sumed c 1, so u is not near the threshold value u=1. of the block becomes zero. As a result, the system is ca- ∼ p In the original variables dX /dt (f f )/√k m, so pable of propagating ultrasonic traveling solitary waves i r s c ∼ − the condition V dX /dt gives in the limit of zero loader plate velocity for any value f i ≪ of accumulated stress above the excitability threshold f f u = 0. The existence of such solutions is a character- V r− s. (29) f ≪ √k m istic feature of excitable systems [1–6]. c In the continuum limit the solution in the form of the This formula shouldin factbe obviousfromthe physical travelingshockwaveisindependentoftheshort-scalebe- considerations. Similarly, one should recover our results havior of the system, except for the precise form of the for the creep-slip models considered in [7] if the charac- dependence of the wave’s velocity on the amount of the teristic velocity of variation of the friction force is small accumulated stress in front of the wave. The latter in enough. fact only weakly depends on the dynamics of the model In contrast, if the condition opposite to the one in atshortlengthscalesif the value ofu issufficiently close Eq. (29) holds, no traveling shock waves can be real- to the slipping threshold u = 1. Note that one might ized in the continuum limit. This can be seen from the 8 following argument. In a traveling wave the profile of u f f =pa2(µ µ ). Onecanseethattheconditionof r s r s should be described by Eq. (12) in which, in the case of Eq−. (29) with V −=10−5 m/s (see [21])is satisfiedif p= f the velocity-weakeningfrictionone shoulddropthe term 2 106 Pa=20 atm, whatgives f f =1.2 10−7 N. r s × − × 2cγu and replace u by u 1 close to the slip point. For this value of p the characteristic sliding velocity will z −There we must have u < 0−, so, according to the mod- be X˙ (f f )/ k m = 1.6 cm/s. The displacement zz r s p ∼ − ified Eq. (12), we will have c < 1 in the traveling wave. ∆X in a single wave can also be estimated as ∆X = Onthe otherhand, the propagationofsubsonicwavesin 2(f f )/k = 1p10−8 m. The latter quantity turns r s p − × the system is impossible since from the dispersion rela- outtobequitesmall. Notethatboth∆X andX˙ increase tion ω2 =1+k2 for the waves in the absence of friction aspincreases. Also,smallerpressurespwouldbeneeded wegetthattheirphasevelocityc=ω/k=√1+k−2 >1 for smaller values of h. From [21] one can calculate α= forallk. Ontheotherhand,numericalsimulationsshow 6.7 10−6 kg/s for this value of p, leading to γ =0.45. that in the discrete models with velocity-weakening fric- In×thepresentpaperwestudiedperfectlyhomogeneous tiontravelingshockwavesareindeedobserved. However, systemwithoutanyexternalnoise. Itisclearthatinreal for ∆x 1 these waves move with speeds very close to systems some degree of noise should be present. It is ≪ the speed of sound and their width is just a few lattice interesting to know what effect the introduction of ran- spacings (unless one is close to the threshold u = 1, see domnesswillhaveonthebehaviorofthetravelingwaves. also[15,16]). Oneshouldthereforeexpectthatinthecon- Here one can distinguish two different situations. If one tinuumlimitonlydiscontinuousshockstravelingwiththe adds some small randomness in the parameters of in- speed of sound are feasible, in contrast to the situation dividual blocks, such as the friction coefficients, spring studiedinthepresentpaper. Thisisacrucialdistinction constants and so forth, one would expect the renormal- of the Burridge-Knopoff model with velocity-weakening ization of the speed of the traveling waves. The other friction and the one with Coulomb friction law (Fig. 2) kind of noise would result in sudden transitions of the studied by us. blocks at rest to sliding. This kind of noise can be read- Aswasalreadydiscussedabove,thepropagationspeed ilyincorporatedintoourfreeboundaryproblembyintro- of the wave depends on the short-scale dynamics of in- ducing generation of pairs of counter-propagating waves dividual blocks. Therefore, in order to be able to per- at random points in space. Here the open question is form numerical simulations of the models under consid- how the frequency of this generation should depend on erationinthe continuumlimit, onehasto useverysmall the distance to the slipping threshold. It is also inter- discretization of time. This should make direct simula- esting how the presence of such a noise would affect the tions of our model rather difficult. On the other hand, wave statistics. we showed that if the initial conditions vary sufficiently slowly,thedynamicsofthesystemreducestothemotion ofindividualshockwaves(seeSec. IV).Therefore,byre- APPENDIX A: formulating the dynamics of the problem in terms of the freeboundaryproblemonecandramaticallyincreasethe Since the variation of u leading to the slip event is of i speed of the simulations, thus being able to observe the order ∆x2 and is small for small ∆x, for the blocks near dynamics for much longer times. This should be impor- theslippointonecanneglectboththevariationofu and i tant in the statistical analysis of our system, especially the term 2γdu /dt in Eq. (7), so in the limit ∆x i in the context of earthquakes (see also [12–14]). − → 0 the right-hand side of this equation simply becomes Let us see how the results obtained from the studies u 1 = const. For convenience, let us introduce the + ofour modelshould translateto realstick-slipproblems. − new variables τ and v , such that i Consider, for example, a Bristol board of thickness h = 2 mm lying on the surface made of the same material τ2 underexternalpressurepandpulledatthetop[21]. The t=ts+τ∆x, ui =u+ u+(∆x)2 vi+ . (A1) − 2 coefficientsµ andµ ofthestaticandtheslidingfriction (cid:18) (cid:19) r s for these surfaces were found to be µr = 0.37 and µs = Since at τ =0 the s-th block just slipped, we must have 0.31, respectively [21]. The memory length which we [see Eq. (10)] should use for the distance a between the blocks was found to be a = 1 µm. A single block in the Burridge- 1 u+ Knopoff model should be associated with a piece of the vs−1(0)= −u , (A2) + board of dimensions a a h, so the mass m of the block should be m = ×2.4 ×10−12 kg, where we used with the initial conditions the density ρ = 1.2 g/cm3×. From the speed of sound ′ v (0)=0, v (0)=0, (A3) c = a k /m [Eq. (3)] we can calculate the values of s s c k and k = k (πa/2h)2 (see the end of Sec. II). Using c pp c where the prime denotes the derivative with respect to c = 4000 m/s, we find k = 3.8 107 N/m, and k = c × p τ. 2.3 101N/m. Thefrictionforcejumppersingleblockis × In the limit ∆x 0 Eq. (7) can be written as → 9 d2v 3 5τ τ2 dτ2i =vi+1+vi−1−2vi, i1arepresentedinFig. (3). Theseresults also show that close to u =0 we have + u 2γ˜u +u=0, (B1) ξξ ξ − 1 c 1+ u2. (A9) with the same initial conditions written in terms of ξ: ≃ 2 + u(0)=u , u (0)=0. (B2) The problem in Eqs. (A3) can be treated analytically + ξ by expanding its solution in the powers of c−1. Indeed, Two situations are possible here. If we have γ˜ > 1, if c is large,for τ

Traveling wave solutions in the Burridge-Knopoff model

Similar Traveling wave solutions in the Burridge-Knopoff model