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ISSN 0145 8752, Moscow University Geology Bulletin, 2011, Vol. 66, No. 1, pp. 13­20. © Allerton Press, Inc., 2011. Original Russian Text © V.S. Zakharov, 2011, published in Vestnik Moskovskogo Universiteta. Geologiya, 2011, No. 1, pp. 15­21.

Models of Seismotectonic Systems with Dry Friction
V. S. Zakharov
Department of Dynamic Geology, Faculty of Geology, Moscow State University, Moscow, Russia e mail: zakharov@dynamo.geol.msu.ru, vszakharov@yandex.ru
Received May 18, 2010

Abstract--Several models of a seismic regime and block dynamics that are based on frictional self oscilla tions in systems with nonlinear dry friction are considered. The set of Burridge Knopoff models is presented to describe the generation of earthquakes with the relative movement of the fault sides. The disk model, which describes rotary dynamics of the crustal blocks, and the model of the block dynamics in the foredeeps, are examined. It is established that a regime of chaotic oscillations exists in a certain range of parameters (fric tional force, the velocity of leading blocks, etc.) in the considered systems. This limits the predictability of these systems. Keywords: seismotectonic systems, dry friction, frictional self oscillations, block dynamics, Burridge Knop off models, deterministic chaos. DOI: 10.3103/S014587521101011X

INTRODUCTION According to contemporary representations, the lithosphere is a hierarchical self similar coarse­dis crete medium. Seismic processes are closely con nected with the structure of the inhomogeneous medium. Understanding of the causes that govern the occurrence of earthquakes and their spatial distribu tion and searching for precursors of earthquakes have always been crucial problems of seismology and seis motectonics. As the information on the spatiotemporal condi tions for tectonic activity and seismicity was accumu lated, the complexity of this process became evident. It is clear that it is governed not only by the structure of the seismotectonic system, its complexity, and a large number of system elements, but also by the char acter of interrelations. In this case, a large part is played by the dissipative forces that organize the func tioning of the entire seismotectonic system. Dissipa tive forces not only dissipate mechanical energy trans forming it to heat, but they can lead to rather complex


dynamics. Dry friction refers to dissipative forces that act in mechanical systems. The purpose of this work is to consider several models of a seismic regime and block dynamics. All of them are based on a self oscillating system with non linear dry friction. THE LAW OF DRY FRICTION The forces of dry friction are forces that appear at the contact of two solid bodies without any fluid or gaseous interlayer between them. The dry frictional force is calculated by the expression Ft = P, where P is the pressing force and = (v) is a friction coef ficient that is determined empirically and which depends on the velocity of the relative movement of friction surfaces. Unlike Newton's viscous friction, the law of dry friction is nonlinear [Basics of Tribiology, 2001]. Fig ure 1 shows the different variants of the quality depen dence of the friction coefficient on the relative veloc ity. The simplest law of dry friction, which was sug
s d

(a)

(b)

(c)

||

||

||

Fig. 1. Dependence of the friction coefficient on the velocity: (a) Amanton's law; (b) Coulomb's law; and (c) the modern models for dry friction.

13


14

ZAKHAROV x
2

(b)

1.7 (a) x1 k m F
1 1

= 3.49; = 2.25 = 1.25

V0 x2 k
2

1.6 1.5 1.4 1.3

kc

m

2

1.2 1.1 1.0 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 x1

1

F

2

Fig. 2. The Burridge and Knopoff model: (a) a two block system; (b) the phase diagram of the system dynamics in coordinates (X1X2) in the regime of chaotic oscillations.

gested by Amanton, has the form shown in Fig. 1a; a more realistic frictional law offered by Coulomb is presented in Fig 1b. The coefficient of static friction s in the Coulomb law is greater than the coefficient of sliding friction d at a relative velocity equal to zero. Further studies indicated that at small values of the relative velocity the friction coefficient decreases as the velocity increases, then it can be constant within a certain velocity range and can grow again at a large rel ative velocity. The law of dependence of the friction force on the velocity of this type is shown in Fig. 1c. In [Marone, 1998] the experimental data on the dependence of the friction coefficient on the velocity of a relative movement is presented. There are many variants of analytical assignment of dry friction fea tures [Dieterrich, 1978; Scholz, Engelder, 1976; Carl son, Langer, 1989; De Sousa Vieira, Herrmann, 1994; Schmittbuhl, et al., 1996]. The dry friction that exists in a system leads to considerable consequences for its dynamics, except for simple dissipation of energy. FRICTIONAL SELF OSCILLATIONS Let us consider the system dynamics that is usually used during the study of friction, both theoretical and experimental ("the load on a moving band"). A load with mass m is attached to the immovable basis by a spring and a damper so that its bottom lies on a hori zontal band that is moving at a constant velocity V0 (the conveyor band). A dry friction force acts between the band and the load. The body is drawn by the band, but the spring inter feres with this forward movement and pulls the body off the band when the elastic force exceeds the fric tional force. The load makes oscillatory movements with a frequency close to the eigen frequency. At a cer

tain moment when the load and the band have the same velocities, the frictional force sharply increases, the load "sticks" to the band and they start moving together until the body is taken off again. This behav ior is called stick­slip. This phenomenon is termed frictional self oscilla tions; these are undesirable for mechanisms and are avoided. But for seismotectonics friction self oscilla tions have a special meaning, since according to mod ern views exactly this type of mechanism causes the onset of earthquakes. We mention that the main feature that leads to the appearance of frictional self oscillations is dry fric tion, i.e., an excess of "static friction" s over sliding friction d and a decrease of the friction force with an increase of the velocity. Here, the specific type of func tion that describes this removal is not important. BURRIDGE­KNOPOFF MODELS A rather simple system under consideration is a variant of the model offered by Burridge and Knopoff [Burridge, Knopoff, 1967]. It is often used as a basis model of earthquake generation during the relative movement of fault sides. The dynamics of the model studied above seems too simple to represent a seismotectonic process; each removal (i.e., model earthquake) occurs in it periodi cally and predictably. However, if a second block is added to the system (Fig. 2a) this leads to a qualitative change in its behavior [Turcotte, 1977]. Here, kc, k1 = k2 = k is the stiffness of springs that model the forces of the elastic connection of blocks between each other and a leading block. F1 and F2 are the friction forces acting on the foundations of the first and the second blocks; 1 and 2 are the respective friction coeffi
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MODELS OF SEISMOTECTONIC SYSTEMS WITH DRY FRICTION (a) (b)

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W



Fig. 3. A three disc system with dry friction: (a) the type of the model; (b) the phase diagram of oscillations of one of the driven discs in a chaotic regime.

cients. This model uses the pure Coulomb law of fric tion for each block s and d (Fig. 1b). The blocks interact in the system as follows: the movement of one block can cause the falling of the other and vice versa. The system dynamics is analyzed in detail in [Turcotte, 1997]. We introduce two govern ing parameters = kc/k (the degree of connection of blocks between each other) and = s2/s1 (the asym metry of static friction for the blocks). The system demonstrates different behaviors depending on the values of these governing parameters. When = 1, periodic self oscillations occur in the system. If the friction coefficients for the blocks are different ( 1) the system behavior becomes more complicated and diverse. Both periodic and chaotic regimes are possi ble in the system depending on the value of the param eter . Figure 2b depicts a phase portrait of the system in the coordinates (X1, X2) in the regime of chaotic oscillations. Thus, we have established that a self oscillating system with dry friction that consists of only two blocks can exhibit complicated chaotic behavior. It is evident that an increase in the number of elements in these systems will lead to even more complicated dynamics. We consider several models of geological systems that consist of several blocks that are con nected by elastic forces and dry friction forces. A THREE DISC MODEL WITH DRY FRICTION Crustal blocks can make both forward and rota tional movements. The interaction across the bound aries of the blocks determines the regional seismicity. To describe the dynamics of a rotational movement, we examine the following three disc model [Vadk ovskii, Zakharov, 2002] that reproduces the main fea tures of behavior of a grain structure. The model is shown in Fig. 3a: one of the discs (the leading disc) moves at a constant angular velocity , two others contact the leading disc and each other. At the points of contact the driven discs are acted on by the moment of dry friction forces (Fig. 1c). In addi tion, the discs are acted on by the moments of forces of
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linear elastic interaction and the moment of forces of viscous resistance. The dynamics of the system under consideration demonstrates various regimes (damped oscillations, regular self oscillations, and chaotic self oscillations) that depend on the model parameters. The governing parameter here is the angular velocity of the leading disc, which models the velocity of a tectonic move ment. At small values of the system reaches the equilibrium state. When the value of increases, a transition to chaotic behavior occurs. Figure 3b shows a phase diagram of the chaotic oscillations of one of the driven discs in the coordi nates of the angle of rotation () is the angular velocity (). In the chaotic regime the system dynam ics exhibits oscillations with different periods and amplitudes, as well as sudden jumps of the velocity of the disc's angular rotation. These jumps represent model "earthquakes." We mention that the values of the jumps are different and are directed along and opposite to the rotation of the leading block. Rare jumps with large amplitudes and multiple jumps with small amplitudes are generated by the same mecha nism. The amplitudes of the jumps and the time inter val between the jumps are distributed chaotically. The precursors of the jumps are not disclosed; this is true for jumps of any amplitude. In other words, one jump forward does not provoke another jump. The sequence of jumps is unique for each initial value. We should mention that the appearance of a chaotic distribution of the moments of jump initiation does not require any random force; it is a consequence of the nonlinear character of the interactions. A simi lar model for lithosphere dynamics that consists of rotating blocks is presented in [Primakov, Shnirman, 1999]. A MODEL OF LINKED BLOCKS MOVING ACROSS A FAULT (THE TRAIN MODEL) The measurements of strains of the Earth's surface indicate the presence of oscillations with different periods and amplitudes. In some places the changes
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ZAKHAROV (a) m k V0 (b)

Time
Fig. 4. The train model of a block moving on a fault: (a) the type of the model; (b) the time diagram of displacements for 50 central blocks.

are relatively smooth, in the others, they are very rapid (earthquakes). The model of blocks linked with each other (the train model, Fig. 4a) is offered to describe the move ment across the fault. Each block is acted on by elastic forces from the adjacent blocks, a viscous force of medium resistance, and a dry friction force from the surface moving at a constant velocity V0 [Vadkovskii, Zakharov, 2002]. Figure 4b shows the time diagram of displacements Xi(t) for 50 central blocks from the total number of blocks in the chain that equals 100. The time increases from left to right. The chain of blocks also exhibits the jumps of velocities of some blocks that cause the adjoining blocks to displace. The moment of time when each jump occurs and the displacement sign are unpredictable. No special behavior precedes a jump. An arbitrary small change in the initial conditions does not lead to a qualitative change in the behavior of the blocks but the sequence of the jumps in time becomes completely different. This model enables us to comprehend the features of seismicity occurrence across a fault, as well as the dynamics of fault sides. The governing parameter in this model is the velocity of the displacement (V0) of the underlying surface.

The above model is actually a variant of the train model by Burridge and Knopoff [Cartwright et al., 1997]; its dynamics also exhibits a chaotic character. Various models that develop the approach of Burridge and Knopoff have been proposed. The book by Tur cotte [1997] considers a two dimensional variant of the model. This system demonstrates the behavior called self organized criticality (SOC) [Bak, Tang, 1989]. The results of its dynamics modeling are widely used to explain the features of a seismic process. A hierarchical modification of the Burridge and Knopoff model [Schmittbuhl et al., 1996] also exists. In this system blocks that interact with each other via elastic connections have sizes that are subject to the power distribution law, and the stiffness values of the con necting springs are proportional to the sizes of the blocks. THE USE OF A DISC MODEL TO DESCRIBE THE ROTATIONAL DYNAMICS OF THE CRUSTAL BLOCKS IN THE AEGEAN­ANATOLIAN REGION The Aegean­Anatolian region is of great interest for geodynamic and seismotectonic investigations. The dynamics of this region are governed by the inter actions between the Euroasian, African, and Arabian
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MODELS OF SEISMOTECTONIC SYSTEMS WITH DRY FRICTION 46

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44

Eurasian plate

42
24 ± 2

40

10 ± 2

38

? ?
30 ± 2

2 9±

36
18 ± 2

1 2 3

34

32 African plate 30 20 25 30 35 Arabian plate 40

4 5 6 45

Fig. 5. A pattern of block dynamics in the Eastern Mediterranean that is based on the GPS data analysis by McClusky, et al. [2000]: 1 is the zone of Caucasus convergence; 2 is the East Turkish distributed strike slip zone; 3 is the Anatolian microplate; 4 is the strike slip and extensional zone in the western part of the North Atlantic fault; 5 is the zone of submeridional extension in Western Turkey and in the northeast of the Aegean plate; 6 is the southwest of the Aegean­Peloponess microplate. The arrows indicate the directions of the horizontal movement, the numbers denote the movement velocity, cm/year; the solid lines represent the faults and boundaries of the blocks; the dashed lines designate the borders of the states.

plates. Qualitatively we can apply a somewhat compli cated model of disc interaction for the description of this complex dynamics. Geological, geodesic, and seismotectonic research has established the complicated block structure of the Eastern Mediterranean. The work by McClusky et al. [2000] presented the data on the horizontal velocity of the GPS points. The analysis of these data makes it possible to reveal the relative rotation of the blocks (microplates) in the region. Especially large move ments occur across the North Atlantic fault. Figure 5 shows the pattern of microplate (block) dynamics that is proposed in McClusky et al. [2000] for the interpre tation of the GPS data. Except for the general dis placement this dynamics causes the appearance of proper movements of the blocks that compose the region. The seismological observations in the Aegean­ Anatolian region show increased seismic activity that is mainly represented by weak earthquakes. The regional seismicity according to the PDE catalog for 1973­2007 is presented in Fig. 6. Almost all earth
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Fig. 6. The seismicity of the Aegean­Anatolian region by the PDE catalog for 1973­2007 and the approximate model of the blocks that are identified during the analysis (shown by the dotted line) according to Simonov et al. [2006]. No. 1 2011


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ZAKHAROV (b)
i i-1 i+1

U0

Fig. 7. A model of block dynamics in the foredeeps: (a) the geodynamic model; (b) the mechanical model.

quakes are small, crusty, with depths not greater than 33 km. The location of epicenters is inhomogeneous: a considerable part of the foci are concentrated in the zones of boundaries between the large blocks (micro plates), such as the North Anatolian fault and the Aegean trough. Zones of concentration of the epicen ters of a lesser scale are also identified. At the same time, between these zones we classify the zones where the frequency of earthquakes is relatively small. Such a situation can result from complex interac tions of the crustal blocks of different scale levels (rank), the general cause (the source of energy) is the movement of the Arabian plate that causes the interac tion and complicated forward and rotational dynamics of relatively stiff large blocks (microplates), such as the Anatolian, Aegean, and others. These large blocks and the zones between them are inhomogeneous and con sist of blocks of a lesser scale that also can make differ ential rotational movements. The interactions across the sides of these different scale blocks are exhibited in the seismicity picture that is observed. We revealed the differential rotational movements of blocks included in microplates in this region based on the analysis of the GPS data in [Simonov, et al., 2006; Zakharov and Simonov, 2010]. The results of the preliminary dynamics are presented in Fig. 6,which shows the identified blocks. The complex structure, dynamics, and relative mobility of the crustal blocks that are determined by the analysis of seismic and geodesic data are also revealed on other time scales. Piper et al. [1997] present the results of paleomagnetic research in this region that point to the relative rotation of these blocks. The system of adjoining blocks models the complex behavior of a so called granular structure, which is a system of blocks with admissible forward and rota tional relative displacements at a discontinuous inflow of energy. Both the differential movements of some elements and cooperative collective movement of agglomerates that temporarily occur and then undergo disintegration is possible in the granular media. THE MODEL OF BLOCK DYNAMICS IN THE FOREDEEPS Areas that adjoin mountain folded structures, for example Ciscaucasia, usually have a complicated lat

eral structure. The geological data show the rather complex dynamics of these zones. Such disintegration can be explained by the collisional interactions of stiff plates; as the ultimate strength is reached the crust (of the upper layers) undergoes brittle failure The subvertical faults divide the upper layer into separate blocks (with a horizontal size of up to several hundreds of kilometers) that are relatively weakly con nected with each other, mainly by friction forces. Being at a state that is close to equilibrium, the blocks "float" on a viscous foundation, viz., the bottom crust. The blocks are connected with a platform region and represent special "keys" (Fig. 7a). The dynamics of these blocks are considerably influenced not only by frictional and buoyancy forces but also by elastic forces that appear when the blocks bend. The region of these blocks that corresponds to the region of submount sedimentary basins is separated from the orogen by a large fault. The rising orogen transfers its vertical movement to the neighboring blocks due to friction forces. They also begin to rise; however, as a result of collective behavior and due to features of frictional forces (their nonlinearity), the character of the movement can be very complicated and significantly different from gradual rising. To study the dynamics of the system behavior, the model in [Koronovsky, Zakharov, 2000; Zakharov, 2001] is offered. Let us consider a linear chain of stiff blocks that are floating in a viscous fluid (Fig. 7b). To take the elastic forces into account, we assume that each block is fastened to a spring (which corresponds to elasticity that characterizes each "key"), so that the spring was not stretched in a state of hydrostatic bal ance. Each block contacts the neighboring lateral sur faces (we will consider them to be vertical), coherence is provided by the friction force. In addition, all the blocks contact and interact with one common leading block via frictional forces; the leading block moves vertically at a certain velocity U0 (which corresponds to the rise of a mountain structure). The dynamics of each block are determined by the balance of the forces acting on it: the gravity force, the buoyancy force (i.e., isostatic forces), elastic forces, frictional forces on the part of the neighboring blocks and the leading block, and viscous forces. Figure 8 shows the time diagram of displacements Xi(t) for a system of nine blocks.
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Displacement

Time
Fig. 8. Time diagrams of the displacements for a system of nine blocks.

It is seen that the system in question is involved in rather complicated oscillating movement, where non periodic oscillations with a specific time that is greater by one order of magnitude appear, along with oscilla tions that are determined by the eigen frequency of each block. The blocks oscillate around one, two, or sometimes three equilibrium positions, the jumps between which are separated by different periods of time. When participating in the common process of processing the incoming energy, some blocks can behave relatively independently, the neighboring blocks can participate in "jumps" in both one way and opposite directions: one block moves upward, the other moves downward. Sometimes several blocks "join together" making joint oscillations, but then their combinations disintegrate. Certain elements of this system have the own "his tory." This model can explain some elements of the dynamics of the submount sedimentary basin. CONCLUSIONS Despite the different temporal­spatial scales, all the systems considered above have the common fea ture of vividly expressed dynamics. They exhibit rather complicated oscillating movements, where nonperi odic oscillations appear along with oscillations that are determined by the eigen frequency of each block. At a certain range of the parameters (the frictional force, the velocity of leading blocks, etc.) the systems consid ered exhibit a regime of chaotic oscillations and sud den jumps between different positions that are sepa rated by different periods of time. When participating in the common process of processing the incoming energy, some blocks can behave relatively indepen
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dently, the neighboring blocks can participate in "jumps" in both the one way and opposite directions. Sometimes the blocks "join together," making joint oscillations, but then their combinations disintegrate. This phenomenon, chaotic behavior, even with the exact assignment of all the parameters, is called deter ministic chaos. A similar situation is observed for a rather broad (but previously unknown) range of parameters. However, any small changes in the veloc ity, the frictional force, etc. leads to the fact that the behavior of every block can change (i.e., certain moments of the jumps become different), but the gen eral regime remains the same in this case. The study of these models makes the complications that researchers encounter during the prediction of earthquakes and other catastrophic events more understandable. Our limited ability to predict the movements of these systems (or the impossibility of making such predictions) is due to their nature. We should emphasize that it is dry friction that plays the key role in the chaotic character of the dynamics of the systems considered here. Thus, mod els of dynamics that take dry friction into account are rather useful for understanding both the features and the complexity of seismotectonic processes and for the Earth sciences in general. REFERENCES
1. Bak, P. and Tang, C., Earthquakes as a Self Organized Critical Phenomenon, J. Geophys. Res., 1989, vol. 94, no. B11, pp. 1535­1537. 2. Burridge, R. and Knopoff, L., Model and Theoretical Seismicity, Bull. Seismol. Soc. of America, 1967, vol. 57, no. 3, pp. 341­371.
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ZAKHAROV 12. Primakov I. and Shnirman M., Type of Trajectory Instability for a Movable Disk Model of the Lithos phere, Phys. of the Earth and Planet. Inter., 1999, vol. 111, pp. 305­315. 13. Schmittbuhl, J., Vilotte J. P., and Roux, S., Velocity Weakening Friction: A Renormalization Approach, J. Geophys. Res., 1996, vol. 101, no. B6, pp. 13911­ 1317. 14. Scholz, C.H. and Engelder, T., The Role of Asperity Indentation and Ploughing in Rock Friction, Inter. J. Rock Mech. Min., 1976, vol. 13, pp. 149­154. 15. Simonov, D.A., Zakharov, V.S., and Lyu, S. Procedures of Analysis of Modern Discrete Movements of Blocks of Geodynamically Active Regions by the GPS Data (by the Example of Aegean­Anatolian Region) in: Oblasti aktivnogo tektonogeneza v sovremennoy drevney istorii Zemli (Areas of Active Tectonogenesis in the Modern Ancient History of the Earth) Proceedings of XXXIX Tectonic Conference, Moscow: GEOS, 2006, vol. 1, pp. 215­219. 16. Turcotte, D.L., Fractals and Chaos in Geology and Geo physics, Cambridge: Cambridge Univ. Press, 1997, sec ond edition, p.398. 17. Vadkovskii, V. N. and Zakharov, V. S., Dynamic Pro cesses in Geology Electronic Handbook, Proceedings of XXXV Tectonic Conference, Moscow, 2002, pp. 86­89. 18. Zakharov, V. S., Model of Block Dynamics in Submon tane Zones, in Sovremennye protsessy geotektoniki (Modern Processes of Geotectonics), Moscow: Nauchny Mir, 2001, pp. 106­109. 19. Zakharov, V. S. and Simonov D. A., Analysis of Modern Discrete Movements of Crustal Blocks of Geodynami cally Active Regions by the GPS Data, Vestn. Mosk. Univ., Ser. 4 Geologiya, 2010, no. 3, pp. 25­31.

3. Carlson, J.M. and Langer, J.S., Mechanical Model of an Earthquake Fault, Phys. Rev. A., 1989, vol. 40, no. 11, pp. 6470­6484. 4. Cartwright, J.H.E., Hernandes Garcia, E., Piro, O., Burridge Knopoff Models as Elastic Excitable Media, Phys. Rev. Lett., 1997, vol. 79, pp. 527­530. 5. De Sousa Vieira, M. and Herrmann, H.J., Self Simi larity of Friction Laws, Phys. Rev. Lett., 1994, vol. 49, pp. 4534­4541. 6. Dieterich, J., Time Dependent Friction and the Mechanics of Stick, Pure Appl. Geophys., 1978, vol. 116, pp. 790­806. 7. Koronovskii, N.V. and Zakharov, V.S., Oscillations of the Crustal Blocks of the Southern Margin of the Scythian Plate (North Ciscaucasia) Caused by the For mation of Forward Bending, Proceedings of XXXIII Tec tonic Conference, Moscow, 2000, pp. 232­235. 8. Marone, C., Laboratory Derived Friction Laws and Their Application to Seismic Faulting, Ann. Rev. Earth Planet. Sci., 1998, vol. 26, pp. 643­696. 9. McClusky, S., Balassallian, S., Barka, A., et al. Global Positioning System Constraints on Plate Kinematics and Dynamics in the Eastern Mediterranean and Cau casus, J. Geophys. Res., 2000, vol. 105, no. B43, pp. 5695­5719. 10. Osnovy tribiologii (treniye, iznos, smazka) (Basics of Tri biology (Friction, Wear, and Lubrication), Chichi nadze, A.V., Ed., Moscow: Mashinostroyeniye, 2001. 11. Piper, J.D.A., Tatar O., and Gursoy, H., Deformational Behavior of Continental Lithosphere Deduced from Block Rotations across the North Anatolian Fault Zone in Turkey, Earth and Planet. Sci. Lett., 1997, vol. 150, nos. 3­4, pp. 191­203.

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