Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 39, 3, pp. 475-505, Warsaw 2001 MODELLING CONTACT PROBLEMS WITH FRICTION IN FAULT MECHANICS Włodzimierz R. Bielski Institute of Geophysics, Polish Academy of Sciences e-mail: wbielski@igf.edu.pl Józef Joachim Telega Institute of Fundamental Technological Research, Polish Academy of Sciences e-mail: jtelega@ippt.gov.pl The aimof this contribution is two-fold.First, we review the frictionmo- dels applied ingeophysics.Thesemodels cover: state- andrate-dependent friction, rate-dependent friction and slip-dependent friction. Second, we propose a new description of friction in the spirit of mo- dern contact mechanics, introducing sliding rules which interrelate the contact stresses with the slip velocity. Sliding rules are formulated in a subdifferential form. Initial-boundary value problems are formulated in the strong and variational forms. By applying Green’s function, the va- riational formulation for finding normal and tangential contact stresses is proposed. Key words: fault mechanics, friction, contact laws, state variables 1. Introduction Frictional behaviour of rocks plays an important role in earthquake pro- cesses and their prediction. Once a fault has been formed, its further motion is controlled by friction, which has a contact property rather than the bulk property. In rock friction studies two aspects are crucial: the stability of engi- neering structures and the mechanics of earthquakes. The modern seismology claims that the earthquakes are processes of cre- ation of discontinuity of displacement fields in upper mantle of the Earth at 476 W.R.Bielski, J.J.Telega differentdepths.Theseprocesses take place on old andnewsystems of tectonic faults. As a rule, many factors play an essential role during processes occuring on tectonic faults: friction, temperature, chemical reactions, phase transitions, porosity, etc., cf Teisseyre (1985, 1995) and Teisseyre andMajewski (2001). Though each of these factors is important, the frictional behaviour of rocks has been investigated in many papers. A program of such investigations can be summarized as follows, cf Rice (1983): (i) characterization of complete behaviour of the slip surface, i.e., finding the distribution of contact (tangent) stresses as a function of normal stress, temperature, slip rate, slip distance and history of slip; (ii) description of mechanical interaction between contact surfaces and the surrounding elastic bodies. Friction is an important phenomenon which has obviously to be taken into account in seismological modelling of material behaviour in a neighbor- hood of tectonic faults, see Ben-Zion and Rice (1995), Dieterich and Kilgore (1994), Rice (1993), Zheng and Rice (1998), Sleep (1997, 1998), Segal and Rice (1995), Rudnicki and Wu (1995), Ranjith and Rice (1999), Senatorski (2000a,b). Contact problemswith friction, including unilateral problems, have been frequently studied in the contactmechanics, cf Bielski andTelega (1994), Brogliato (1999), Telega (1988) and the references cited therein. The investigation of friction laws on geological faults emerges as a key is- sue for earthquake modelling. Two types of processes have to be taken into account: quasi-static one as a long term process of slow loading and duration of about a few scores of years, and the second one, the rapid unloading, i.e. a dynamic process occuring during a few seconds. Both the dynamic and quasi- static processes are necessary to complete the description of the phenomenon of friction on tectonic faults. It is convenient to distinguish three models of frictional sliding studied in geophysical literature. The first one is themodel of rate- and state-dependent friction. In this theory, the so-called state variables are applied and the idea is due to Rabinowicz (1965). Next it has been developed by Dieterich (1978, 1979), Ruina (1980, 1983), Rice (1983), Scholz (1994, 1996, 1998), cf also the references therein. The secondmodel is the rate-dependent friction, cfMadariaga et al. (1998), Cochard andMadariaga (1994). The third model considered in seismology is the slip-dependent friction, cf Favreau et al. (1999), Campillo and Ionescu (1997), Descalu et al. (2000), Ionescu and Campillo (1999), Ionescu and Paumier (1997). Modelling contact problems... 477 In this contribution we shall first briefly describe general aspects of tecto- nic plates and provide classification of tectonic faults sufficient for this study. Next, we review the role of state variables inmodelling the friction on tectonic faults. Following our earlier contribution (Bielski and Telega, 2000), a more elaborate description of the sliding process with friction which includes not only the friction condition but also the sliding rule is introduced. The sliding rules are formulated in the subdifferential form, convenient for variational for- mulations. In the general case both the friction condition and the sliding rule dependon state variables. Finally, variational formulationwill be proposed for a quasi-static problem modelling geological faults. For the sake of simplicity, it is assumed that contacting tectonic plates are made of anisotropic, linear elastic materials. Physically more involved material behaviour can likewise be considered; for instance, inRowshandel andNemat-Nasser (1986), the founda- tion is viscoelastic. Theproblemof anisotropic fault regionwas also considered by Rybicki (1992). This paper is confined to quasi-static problems for contacting tectonic pla- tes. However, amore general than the usually used rate- and state-dependent model of friction is taken into account. Also, anisotropy and inhomogeneity of tectonic plates are included in our approach in a natural manner. 2. General aspects of tectonic plates Plate tectonics is a model in which the outer shell of the earth is divided into a number of thin, rigid plates that are in relative motion with respect to one another. The relative velocities of the plates are of the order of a few scores of milimeters per year, for instance for the San Andreas fault the ave- rage velocity is 0.5-4.5 mm/year. A large number of all earthquakes, volcanic eruptions, andmountain formingoccurs at plate boundaries. Figure 1 sketches the distribution of themajor surface plates. The plates aremade up of relati- vely cool rocks and have an average thickness of about 100km.The plates are being continually created and absorbed. At ocean ridges, the adjacent plates diverge from each other in a process known as seafloor spreading. As the adjacent plates diverge, hot mantle rock ascends to fill the gap. The hot, solidmantle rock behaves like a fluid because of solid-state creep process.As the hotmantle rock cools, it becomes rigid and accretes to the plates, creating new plate area. For this reason, ocean ridges are also known as accreting plate boundaries. 478 W.R.Bielski, J.J.Telega Fig. 1. Themap of main tectonic plates and their boundaries, after Turcotte and Schubert (1982) Modelling contact problems... 479 Fig. 2. Schematic presentation of subduction processes, after Press and Siever (1986) The main oceanic ridges are depicted in Fig.1. Because the surface area of the earth is essentially constant, there must be a complementary process of plate consumption. This occurs at ocean trenches. The surface plates bend anddescend into the interior of the earth in aprocess knownas subduction.At an ocean trench, the two adjacent plates converge, and one descends beneath the other. For this reason ocean trenches are also known as convergent plate boundaries. The distribution of the trenches is depicted in Fig.1 by triangular symbols, which indicate the direction of subduction. A cross-sectional view of the creation and consumption of a typical plate is illustrated in Fig.2. The part of earth’s interior that comprises the plates is referred to as the litho- sphere. The rocks that make up the lithosphere are relatively cool and rigid; as a result, the interiors of the plates do not deform significantly as theymove about the surface of the earth.When the platesmove away fromocean ridges, they cool and thicken. The solid rocks beneath the lithosphere are sufficiently hot to be able to deform freely; these rocks comprise the asthenosphere, which lies below the lithosphere. The lithosphere slides over the asthenosphere with relatively little resistance.As the rocks of the lithospherebecomes cooler, their density increases because of thermal contraction. As a result, the lithosphere becomes gravitationally unstable with respect to the hot asthenosphere be- neath. At the ocean trench the lithosphere bends and sinks into the interior of the earth because of this negative buoyancy. Major faults separate the de- scending lithospheres from the adjacent overlying lithospheres. These faults are the sites of a large number of the great earthquakes. Examples are the Chilean, Alaskian, San Francisco (San Andreas Fault), Anatolian Fault ear- thquakes, as well as Chinese and Japan faults. The location of the descending lithospheres can be accurately determined by the earthquakes occuring in the cold, brittle rocks of the lithosphere. Earthquake source dynamics provides the 480 W.R.Bielski, J.J.Telega key elements for the prediction of strong ground motion and to understand the physics of earthquake initiation, propagation and healing. Recent studies indicate the fundamental role of friction in earthquakes, cf Cotton and Cam- pillo (1995), Beroza andMikumo (1996), Ide andTakeo (1997), Fukuyamaand Madariaga (1998), Iio (1997), Lockner (1998), King and Cocco (2001), Lapu- sta et al. (2000), Ben-Zion and Rice (1997), Cochard and Rice (2000), Brown (1998), Beeler et al. (1996), Boatwright and Cocco (1996), Place and Mora (1999), Oglesby et al. (2000), Nielsen et al. (2000), Richardson and Marone (1999), Roy andMarone (1996), Sleep (1995, 1999). 3. Classification of faults Threre is an abundant classification of tectonic faults from the geological point of view. For our purposes are sufficient the following classifications of faults, followed by Turcotte and Schubert (1982). One may distinguish three main types of faults. Every other fault can be treated as a combination of the three main types. In general, a certain characteristic type dominates in each fault. (i) As the first type consider the thrust faulting. Thrust faultings occur when the oceanic lithosphere is thrust under the adjacent continental (or oce- anic) lithosphere at an oceanic trench. Thrust faults also play an important role in the compression of the lithosphere during continental collisions. Ide- alized thrust fault is depicted in Fig.3. The elevating block is known as the hanging wall, and the depressed block is called the foot wall. The upward movement of the hanging wall is also referred to as reverse faulting. Let the stresses in the x, y, and z directions be the principal stresses (x, z are the horizontal coordinates, y is the vertical coordinate). The vertical component of the stress σyy is the overburden or lithostatic pressure σyy = ̺gy. The vertical deviatoric stress σDyy is zero. To produce the thrust faults, a compressive deviatoric stress applied in the x direction σDxx is required, σDxx > 0. The horizontal compressive stress is σxx = ̺gy+σ D xx, therefore it exceeds the vertical lithostatic stress or σxx > σyy. For the fault geometry shown in Fig.3 it is appropriate to assume that there is no strain in the z direction. In this particular situationwe canwrite σDzz = νσ D xx. The deviatoric stress in the z direction is also compressive, but its magnitude is a factor of ν times less than the deviatoric applied stress. Therefore the horizontal Modelling contact problems... 481 Fig. 3. Schemat of three main types of tectonic faults, after Press and Siever (1986) compressive stress σzz = ̺gy+σ D zz = ̺gy+νσ D xx exceeds the vertical stress σyy, but it is smaller than the horizontal stress σxx. Thrust faults satisfy the condition σxx >σzz >σyy. The vertical stress is the least compressive stress. (ii)Normal faulting accommodates horizontal extensional strain. It occurs on the flanks of oceanic ridges where new lithosphere is being created. Normal fault also occurs in continental rift valleys where the lithosphere is being stret- ched. Applied tensile stresses can produce normal faults as shown in Fig.3. The displacements on the fault planes dipping at an angle to the horizontal lead to horizontal tensile strain. Normal faulting is associated with a state of stress in which the vertical component of stress is the lithostatic pressure σyy = ̺gy and the applied deviatoric horizontal stress σ D xx is tensile σ D xx < 0. The horizontal stress σxx = ̺gy+νσ D xx is therefore smaller than the vertical stress σyy σyy >σxx Consequently, deviatoric stress in the z direction σDzz is also tensile, but its magnitude is a factor of ν smaller than the deviatoric stresses applied. The total stress σzz = ̺gy+ νσ D xx is smaller than σyy but is larger than σxx. Normal faults satisfy the condition σyy >σzz >σxx, thus the vertical stress is the maximum compressive stress. Both thrust faults and normal faults are 482 W.R.Bielski, J.J.Telega also known as dip-slip faults because the displacement along the fault takes place on a dipping plane. (iii) A strike-slip fault is a fault along which the displacement is strictly horizontal. Thus there is no strain in the y direction. The state of stress in the strike-slip faulting consists of a vertical litho- static stress σyy = ̺gy and horizontal deviatoric principal stress that are compressive in one direction and tensile in the other. One horizontal stress will thus be larger than σyy while the other will be smaller, so we have σxx >σyy >σzz or σzz >σyy >σxx For the strike-slip faulting the vertical stress is always the intermediate stress. 4. Friction laws involving internal parameters Brace and Byerlee (1966) hypothesized that stick-slip instabilities in the observed laboratory friction experimentsmight stand for a goodmodel to ear- thquake rupture.Consequently, laboratory experiments are thought asmodels of possible fault motion in the earth. Experiments have been performed with many rock types, with and wi- thout various fault gouge layers, at a range of slip rates, confining pressure, pore pressures, temperatures, and in machines with different geometry and compliances, cf Blanpied et al. (1998), Dieterich (1978, 1979), Dieterich and Conrad (1984), Jaeger andCook (1976), Mair andMarone (1999), Morrow et al. (2000), Olsen et al. (1998), Savage et al. (1996), Sleep (1999), Weeks and Tullis (1985). A different approach to friction experiments consists in postulating a con- stitutive description of a surface slip from which earthquake or laboratory experiments can be predicted throughmodelling, cf Ruina (1980, 1983), Rice (1983,1993), Scholz (1994, 1996, 1998), Segal and Rice (1995), Sleep (1995, 1997, 1998), Zheng and Rice (1998), Ben-Zion and Rice (1995, 1997), Cao and Aki (1986). Such modeling of elastic systems reveals that instabilities in frictional slip depend on a reduction of the friction force during some part of the sliding, i.e. on slip weakening. For this reason, the role of slip weakening has been investigated in many papers. Particularly, Byerlee (1970) suggested that the friction coefficient varies frompoint to point on slip surfaces and that Modelling contact problems... 483 instabilities are associated with decrease in the friction force from its peak values as sliding proceeds. Dieterich (1972) claims that slip weakening occurs after a time-dependent healing during stationary contact. Similar mechanisms have been earlier pro- posed as a basis for slip instabilities, primarily in metals, Rabinowicz (1965). Basing on the ideas of Rabinowicz (1965), Dieterich (1978,1979) and Ruina (1980) studied and developed a class of friction laws based on using the state variables. Ruina (1983) exploited the experimental data by Dieterich (1979) and proposed a model of friction involving state variables. This author provided examples to characterize the state variables and to study the stability of steady sliding, neglecting the inertia forces. Also a friction law based on one state variable was used. Let us pass to a brief presentation of the Ruina (1983) model. This model comprisesbasic experimentally observed features, especially the followingones: fadingmemory and steady-state, positive instantaneous slip rate-dependence, and negative dependence on the recent slip rates. These ideas and observations led to the following description of friction. Let τ be the shear stress and σ let denote the normal stress. After Ruina (1983) we write τ =σF(ϑ,V ) (4.1) where ϑ is a state variable (or a collection of such variables, ϑ = (ϑi), i=1, . . . ,n), V is the rate of the slip. The evolution equation for ϑ has the form dϑi dt =Gi(σ,V,ϑi) (4.2) From the practical viewpoint, the number of state variables ϑi should be small. Thevariables ϑi then represent somekind of average of anundoubtedly complicated surface state. The temperature of the surface can be taken as a single state variable if the heat flow is idealized as being dependent only on the temperature of the surface and the temperature of an external constant temperature reservoir. Detailed analysis of experiments made on different types of rocks lead to the following description of friction provided that one state variable is used τ =σ ( µ0+ϑ+A ln V Vc ) (4.3) ϑ̇=− V dc ( ϑ+B ln V Vc ) 484 W.R.Bielski, J.J.Telega Here µ0 is the coefficient of static friction, A and B are constitutive pa- rameters to be determined by experiments, and dc is the characteristic slip distance depending on the surface. This law is valid for a large range of slip rates and shares the apparent defect of no-healing (no change of ϑ) for a zero slip rate. This law can be illustrated graphically as in Fig.4. Fig. 4. Friction stress at constant normal stress versus slip rate (of ln), after Ruina (1983) In Figure 4 the lines of constant state, ϑ, are light solid lines and show the instantaneous positive dependence of τ on the slip rate V . The heavy line is the steady state friction law and is a decreasing function of the slip rate in the example of Fig.4 (B>A). As governed by Eq. (4.3)2, ϑ decreases above the steady state line, below it ϑ increases. Any slip corresponds to a penmotion on the plot of Fig.4 and is the simultaneous solution of the friction law and any constraints imposed by the loading mechanism. The arrows indicate the component of this motion perpendicular to the lines of constant ϑ. Ruina (1980) derived an experimentwhich cannot be described by a single state variable law of the form (4.1), because of violation of condition (4.2). He showed that his experiment is well described by a friction law involving two state variables τ =σ ( µ0+ϑ1+ϑ2+A ln V Vc ) (4.4) ϑi =− V di ( ϑi+B ln V Vc ) i=1,2 This model is also called the Ruina-Dieterich model, cf Perrin et al. (1995). Modelling contact problems... 485 The model is referred to as the slip model since the state evolves only when there V 6=0. As previously, V is the one-dimensional slip velocity. The quantities A,B, Vc, and di (i=1,2) are constants to be determined by experiment, di is the slip length scale for state evolution; A and B, both positive, account for the short-time velocity strengthening and for the steady- state velocity weakening, respectively. The model specified by Eqs (4.4) can be extended to arbitrary number of state variables, ϑi, i = 1,2, . . . ,n, each of them having specific weakening constant Bi and length scale di. However, due to its simplicity, most frequently used is the model with one or two state variables, cf Ruina (1980, 1983), Weeks and Tullis (1985). Now we describe the Dieterich-Ruina slowness model, cf Perrin et al. (1995). This model has the following form τ(t)=σ [ µ0−A ln ( 1+ V∞ V (t) ) +B ln ( 1+ ϑ(t) ϑ0 )] (4.5) dϑ(t) dt =1−ϑ V (t) L One might think of the state variable ϑ here in an abstract way, cf Ruina (1980, 1983) and (4.1). Dieterich (1979) and Dieterich and Conrad (1984) interpret it as the average age of the load supporting the contacts between the sliding surface. In that case the constitutive law of the form (4.5) is more sensible than the one of the form (4.3)2, since it yields dϑ dt =1 for V =0 That contact time interpretation led Dieterich to use extensively equations (4.5), although equation (4.5)2 seems to have been written first by Ruina (1980). The quantities τ0,A,B,V∞, and ϑ0 are cut-offs for high velocity and short contact duration. Perrin et al. (1995) used a regularized version of Dieterich-Ruina model (4.5) to study the self-healing slip pulse on a frictional surface. Dieterich (1992) pointed out that the model presented by (4.3) leads to non-physical behaviour for extremely low slip velocities. The same happens for long ”con- tact times” ϑ. To remedy these drawbacks, Perrin et al. (1995) introduced two cut-off velocities V0 and V1 andmodified themodel (4.5) as follows τ(t)=σ [ µ0+A ln ( V0+V (t) V∞+V (t) ) +B ln ( 1+ ϑ(t) ϑ0 )] (4.6) dϑ(t) dt =1−ϑ V1+V (t) L 486 W.R.Bielski, J.J.Telega Notice that the state variable ϑ is contained in [0,L/V1]. This might be illogical if ϑ had to be interpreted as the contact time. However, considering a cut-off preciselymeans thatwe are getting outside themeasurable range and it is by nature artificial. Chester (1994) extendedRuina’s friction law and included the temperatu- re. In the case of one state variable, the friction coefficient µ is then expressed by µ=µ0+A [ ln V Vc + QA R (1 T − 1 T∗ )] +Bϑ (4.7) Here QA is the apparent activation enthalpy, T is the absolute temperature, and R denotes the gas constant. Obviously T∗ is a reference temperature, such that µ evolves toward µ0 when V = Vc and T = T∗. The evolution equation for the state variable is modified to the form ϑ̇=− V dc [ ϑ+ln (V Vc )+ QB R (1 T − 1 T∗ )] (4.8) The apparent activation enthalpies, QA and QB, presumably reflect the rate-limiting steps in processes responsible for the direct and evolution effects, respectively. For the steady state, i.e. if ϑ̇=0, then µss =µ0+(A−B)ln V Vc + AQA−BQB R (1 T − 1 T∗ ) (4.9) Here µss denotes the coefficient of friction for the steady state. We observe that for T∗ = T , Eqs (4.7) and (4.8) reduce to Ruina’s equations Eqs (4.3). Some results concerning the temperature-dependent friction are depicted in Fig.5-Fig.8. 4.1. Slip-dependent friction Up to now we dealt with sliding in one direction only. Let us pass to the general case. Let Ω ⊂ R3 be a sufficiently regular domain and Γ = ∂Ω its boundary. Γ consists of three nonoverlapping parts: Γ0, Γ1, and Γc, such that Γ = Γ0 ∪Γ1 ∪Γc and the surface measure of Γc is positive. The bar over a set denotes its closure. Γc is the surface of possible contact, for instance the fault surface. By N = (Ni) we denote a unit exterior vector normal to Γc. Latin indices run from 1 to 3 and the summation convention is used throughout the paper. A vector v=(vi) defined on Γ may be decomposed as follows v= vNN+vT (4.10) Modelling contact problems... 487 Fig. 5. Systematic of the friction parameters (A-B). (a) Dependence of (A-B) on temperature for granite. (b) Dependence of (A-B) on pressure for granular granite. This effect, due tu lithification, should be augmented with temperature, after Scholz (1998) Fig. 6. Friction of graphite powder along the inclined interface at a constant confining pressure of 60ṀPa during velocity and temperature stopping. Velocity and temperature stepping sequence is shown, simplified after Chester (1994) 488 W.R.Bielski, J.J.Telega Fig. 7. Representative results of velocity and temperature stepping experiment on quartz gouge. Velocity and temperature stepping is shown, simplified after Chester (1994) where vN = viNi denotes the normal component of v, while vTi = vi−vNNi are its tangential components. If σ = (σij) is the stress tensor, a similar decomposition holds for the stress vector (σijNj) defined on the boundary Γ . Thus we write σijNj =σNNi+σTi (4.11) where σN =σijNiNj and σTi =σijNj −σNNi. The slip-dependent friction in quasi-static and dynamic cases was conside- red in a series of papers by Ionescu andPaumier (1997), Ionescu andCampillo (1996), Favreau et al. (1999), Campillo et al. (1996). These authors considered the contact problemswith friction between a linear elastic bodyand a rigid fo- undation. The elastic body is an infinite elastic strip bounded by two planes. Such a strip is in contact with the rigid foundation and submitted to she- aring, or the half-spaces being in contact. Quasi-static and dynamic stick-slip motions are related to the earthquake instabilities. On the contact interface the friction, the Coulomb law with a slip-dependent friction coefficient was used provided that normal pressure was prescribed, see Ionescu and Paumier Modelling contact problems... 489 Fig. 8. Comparison of model simulations and detrended friction records from velocity and temperature stepping experiments on quartz gouge under (a) dry and (b) water saturated conditions. Friction versus shear displacement fromsimulations shown by heavy line is superposedwith friction recordf from experiment. The velocity and temperature stepping sequence is shown, after Chester (1994) (1997). Only the anti-plane problemwas studied, both the static and dynamic cases. Let us consider the shearing of an infinite elastic slab bounded by two planes: x1 = l, x1 =0, and x2 =h, x2 =0. On the contact surface Γc = [0, l]×{h}×R, the slab is in contact with friction with the rigid body which pushes it with the constant normal force σ=σ22 =−S or σ(u)N ·N =−S on Γ1 where u is the displacement field, σ = σ(u) is the stress tensor and N is the outward unit normal vector. Along Γ0 = [0, l]×{0}×R the displacement is prescribed u1 =0 u2 =0 u3 =B 490 W.R.Bielski, J.J.Telega and on Γc = {0, l}× [0,h]×R u1 =0 σ12 =σ13 =0 Let uA =0 and u2 =u2(x2) ∂u3 ∂x3 =0 Since no perturbation of the equilibrium in the x1-direction is considered, we get u2(x2)=− S λ+2G x2 (4.12) where λ,G > 0 are the Lamé constants. Let us denote by Ω the rectangle Ω := (0, l)× (0,h); moreover we set w :=u3−B ( 1− x2 h ) First, we describe the static case. In this case the slip-dependent friction law on Γc is described by σT(u)=−Sµ(|uT |) uT |uT | if uT 6=0 on Γc (4.13) and |σT(u)| ¬µ(0)S if uT =0 on Γc (4.14) Here uT and σT are the tangential displacement and tangential stress, re- spectively. The equlibrium equation divσ=0 (4.15) and the boundary conditions lead to the following problem: Find w : Ω→R such that ∆w=0 in Ω ∂w(x1,x2) ∂x1 =0 for x1 = l and x1 =0 ∀x2 ∈ (0,h) w(x1,0)= 0 ∀x1 ∈ (0, l) (4.16) G ∂w(x1,h) ∂x2 +µ(|w(x1,h)|)S sgnw(x,h) = q if w(x1,h) 6=0 ∣∣∣G ∂w(x1,h) ∂x1 −q ∣∣∣¬µ(0)S if w(x1,h)= 0 Modelling contact problems... 491 where q is the tangential stress, which corresponds to the stick case, i.e., q=GB/h. In the dynamic case, the slip-dependent friction law on the contact surface is described by the following system σT(u)=−Sµ(|uT |) ∂u ∂t ∣∣∣ ∂u ∂t ∣∣∣ −1 if u̇T 6=0 on Γc |σT(u)|=−Sµ(|uT |) if u̇T =0 on Γc Themomentum balance law divσ= ̺ü and the boundary conditions lead to the following dynamic problem: Find w : R+×Ω→ R such that ẅ(t)= c2∆w(t) in Ω ∂w(t, l,x2) ∂x1 = ∂w(t,0,x2) ∂x2 =w(t,x1,0)= 0 G ∂w(t,x1,h) ∂x2 +µ(|w(t,x1,h)|)S sgn ( ẇ(t,x1,h) ) = q if ẇ 6=0 ∣∣∣G ∂w(t,x1,h) ∂x2 −q ∣∣∣¬µ(|w(t,x1,h)|)S if ẇ(t,x1,h) 6=0 w(0)=w0 ẇ(0)=w1 in Ω Here c= √ G/̺ is the shear velocity and w0,w1 are the initial conditions. The static analysis of the first of the formulated problems was performed by Ionescu and Paumier (1997) using variational methods. 5. Friction conditions and sliding rules The descriptions of friction on geological faults discussed previously are confined to one-dimensionalmodelling of the change of the friction coefficient. In this section we propose an alternative and rather general approach to mo- delling the friction condition and sliding rule in the spirit of modern contact mechanics, cf Telega (1988). In the the comprehensive rewiev paper (Shillor et al., 2002), the available variational and numerical methods of solving qu- asistatic contact problems are discussed. Let Ωa (a = 1,2) be a domain in the three-dimensional physical space occupied by a linear-elastic body in its undeformed state. Γc denotes the contact surface (the fault surface) of the two contacting bodies.Unboundeddomains are not excluded. Let Na =(Nai ) 492 W.R.Bielski, J.J.Telega denote the outer unit normal vector to ∂Ωa. We set N =N2 =−N1. Let σ a = (σaij) (i,j =1,2,3) be the stress tensor in the body Ω a; ua stands for the displacement vector. Moreover, by σaN =σ a ijN a i N a j we denote the normal component of the stress vector and σaT = (σ a ijN a j )− σ a NN a is the tangent stress vector, while [[uT ]] =u 1 T −u 2 T denotes the jump of the tangent displa- cement across the fault surface Γc. Throughout this paper the summation convention is consequently applied, except that a = 1,2. According to the action-reaction principle we set σT = σ 1 T = −σ 2 T . In the absence of state variables, the friction condition is assumed to be given by f(σN,σT) ¬ 0, where f is a continuous function. Anisotropic friction is not precluded. For a fixed σN we introduce a set K(σN) of admissible tangential stresses as follows K(σN)= {τ| f(σN,τ)¬ 0, τ ·N =0 on Γc} Prior to the formulation of the friction law we recall the definition of a sub- differential of a convex function. If f : Rn → R is a convex function then its subdifferential at x0 is a subset of R n such that ∂f(x0)= {y∈ R n : f(x)−f(x0)­〈y,x−x0〉 ∀x∈ R n} Here 〈y,x〉 = yixi. For more details the reader is referred to Rockafellar (1970). We assume that K(σN) is convex and closed while the sliding rule has the subdifferential form [[u̇T ]]∈ ∂IK(σN)(σT) (5.1) where u̇T = ∂u/∂t and IK(σN) is the indicator function of K(σN), i.e. IK(σN)(τ)=    0 if τ ∈K(σN) ∞ otherwise As usual, ∂IK(σN) stands for the subdifferential of the function IK(σN). In the variational formulation given in the next section, the frictional dissipation density will be involved. It is determined by D(σN, [[u̇T ]]) = sup{[[u̇]] ·τ | τ ∈K(σN)} (5.2) Obviously, D(σN, [[u̇T ]]) = [[u̇T ]] ·σT . Our approach includes anisotropic fric- tion. In the case of the classical Coulomb friction condition we have D(σN, [[u̇T ]]) =µ|σN| · |[[u̇T ]]| Modelling contact problems... 493 Then (5.1) takes the following form [[u̇T ]] =−λσT λ­ 0 (5.3) If the state variables ϑp (p=1, . . . ,n) are employed for the description of friction on the fault then the friction condition is assumed in the form f(σN,σT ,ϑp)¬ 0 (5.4) For fixed σN and ϑp, p=1, . . . ,n, the set of admissible tangential stresses is given by K1(σN,ϑp)= {τ | f(σN,τ,ϑp)¬ 0, τ ·N =0 on Γc} (5.5) In this case the sliding rule may also be assumed in the subdifferential form σT ∈ ∂3D(σN,ϑp, [[u̇T ]]) p=1, . . . ,n (5.6) to which the evolution equation for ϑp should be appended ϑ̇p =Hp(t,σN,ϑm, [[u̇T ]]) m,p=1, . . . ,n (5.7) Here ∂3D(σN,ϑp, [[u̇T ]]) denotes the subdifferential of the frictional dissipa- tion density with respect to the third variable. Particularly, suppose that the friction condition is given by f(σN,σT ,ϑp) = |σT |−µ(ϑp)σN ¬ 0. Then we have D(σN,ϑp, [[u̇T ]]) = |σT ||[[u̇T ]]|=µ(ϑp)|σN|[[u̇T ]]| (5.8) We conclude that the friction coefficient may depend on the slip velocity via the state variables. Remark 5.1. It may happen that the friction condition does not depend on the normal stress σN. Specific case is provided by the friction condi- tion used by Cochard andMadariaga (1994). These authors employ the following velocity-dependent condition in the case of antiplane shear, cf Section 6 below f(σyz, [[u̇]]) = |σyz|−σ 0 yz V0 V0+[[u̇]] (5.9) where V0 is a reference velocity that determines the rate of slip velocity weakening and σ0yz is the maximum traction drop, reached when the slip velocity is very large. The friction condition can be obtained from 494 W.R.Bielski, J.J.Telega the condition depending on an internal variable ϑ. More precisely, let the condition f̃ depend on σyz and ϑ, where dϑ dt =H([[u̇]]) ϑ(0)=ϑ0 (5.10) Solving the last equation we get ϑ=h([[u̇]]) (5.11) Substituting (5.10) into f̃(σyz,ϑ) we obtain f(σyz, [[u̇]]) = f̃ ( σyz,h([[u̇]]) ) (5.12) Remark 5.2. Friction conditions may possess no convexity property. Then the subdifferential ∂ should be replaced by the generalized subdifferen- tial ∂, cf Panagiotopoulos (1993). Instead of variational inequalities we have to deal then with so-called hemivariational inequalities. 6. Classical and variational formulations of the fault contact problem Now we pass to the formulation of a quasi-static initial-boundary value problem in the presence of a fault. The fault is treated as a contact surface or interface between two anisotropic, linear-elastic bodies. It can also be mo- delled as a closed crack in the elastic body. The interface is modelled by the subdifferential sliding rule (5.6).We set ∂Ωa =Γ a 0∪Γ a 1∪Γ a 2, Γc =Γ 1 2 =Γ 2 2 and formulate the quasi-static contact problem. Problem (P) Find ua(x,t) (a=1,2) and ϑp(x,t) (p=1, . . . ,n), such that σaij,j(u a)+Bai =0 in Ω a× [0,T ] σaij(u a)= aaijklεkl(u a) in Ωa× [0,T ] u a(x,t)= 0 on Γa0 × [0,T ] σaijN a j =F a i on Γ a 1 × [0,T ] σT ∈ ∂3D(σN,ϑm, [[u̇T ]]) on Γc× (0,T) Modelling contact problems... 495 ϑ̇p =Hp(t,σN,ϑm, [[u̇T ]]) on Γc× (0,T) m=1, . . . ,n u a(x,0)=ua0(x) for x∈Ω a ϑp(x,0)=ϑ 0 p(x) for x∈Γc where εkl(u) = (uk,l+ul,k)/2, and B a and Fa are the applied body forces and surface tractions, respectively. The functions Hp, u a 0 and ϑ 0 p are given. The formula σaij(u a)= aaijklεkl(u a) expresses the anisotropic Hooke’s law. To obtain the variational formulation we set aa(ua,va)= ∫ Ωa aaijklεij(u a)εkl(v a) dx a(u,v)= 2∑ a=1 aa(ua,va) (6.1) L(v)=L(v1,v2)= 2∑ a=1 (∫ Ωa Baiv a i dx+ ∫ Γa 1 Fai v a i ds ) where u = (u1,u2), v = (v1,v2). It can readily be shown that the problem (P) may be transformed to the variational formulation. Problem (Pv) Find ua =ua(x,t), x∈Ωa (a=1,2), t∈ [0,T ] and ϑp (p=1, . . . ,n), such that ua(x,0)=ua0 (x∈Ω a), ϑp(x,0)=ϑ 0 p(x) (x∈Γc) and a(u,v− u̇)+ ∫ Γc D(σN,ϑp, [[vT ]]) dΓ − ∫ Γc D(σN,ϑp, [[u̇T ]]) dΓ ­L(v− u̇) (6.2) ∫ Γc [ ϑ̇p−Hp(σN,ϑm, [[u̇T ]] ] ηp(x) dΓ =0 for all test functions v= v(x), ηp = ηp(x). The inequality (6.2)1 provides an example of an implicit variational inequality; more precisely, it is a variational inequality of the second kind. From the physical point of view, it represents the principle of virtual velocities in the presence of friction. The variational formulation proposed can be used for the derivation of numerical procedures, cf Johansson (1992) and Section 7. 496 W.R.Bielski, J.J.Telega 7. Green’s function and contact stresses Starting from the problem (Pv) we can derive the so-called dual formu- lation for the determination of stresses σN and σT on the interface, i.e., on the fault. Applying the procedure developed by the second author, see Telega (1991), we arrive at the dual problem. Problem (PG) Find Σ =(σijnj) = (σN(x,t),σT(x,t)), σT ∈K1(σN,ϑp), and ϑp(x,t), x∈Γc, such that for all t∈ (0,T) such that ∫ Γc 〈 sT(x)−σT(x,t), d dt [[[û(x,t)]]T +(G 1+G2)Σ(x,t)]T 〉 dΓ(x)­ 0 ∫ Γc ( [[û(x,t)]]N +[(G 1+G2)Σ(x,t)]N ) ϕ(x) dΓ(x)= 0 (7.1) ∫ Γc ( ϑ̇p(x,t)−Hp(σN,ϑm, [[u̇T ]]) ) ηp(x) dΓ(x)= 0 ϑp(x,0)=ϑ 0 p(x) Σ(x,0)=Σ0(x) x∈Γc for all S = (sN,sT), sT ∈ K1(σN,ϑp) and for all sufficiently regular ϕ. Similarly to Section 4, 〈·, ·〉 denotes the scalar product in R3. Obviously Ga (a=1,2) denotes the Green function for the domain Ωa; moreover ûak(x,t)= ∫ Ωa Bai (y,t)G a ik(x,y)dΩ a(y)+ ∫ Γa 1 Fai (y,t)G a ik(x,y)dΓ(y). (7.2) In Eqs (7.1)1,2 the following notation is used [(G1+G2)Σ]i(x,t) = ∫ Γc [G1ij(x,ξ)+G 2 ij(x,ξ)]Σj(ξ,t) dΓ(ξ) We observe that the dual problem (PG) enables us to find the normal and tangential stresses on the fault surface Γc. The inequality (7.1)1 is a quasi- variational inequality since the set of constraints K1 depends on the solution. Remark 7.1. Dual formulation for the static problemwith friction was exa- mined in Bielski and Telega (1985). Modelling contact problems... 497 8. Antiplane deformation In order to illustrate the approach developed in the previous section, we consider the antiplane crack problem in an isotropic infinite space. Now Ω = R3 and the contact surface, i.e. the crack Γc, is defined by Γc = { (x,y,z)∈ R3 | −ℓ¬x¬ ℓ, −∞ 0. The displacement vector field is assumed to be continuous on Ω\Γc. In the case of the antiplane deformation we have, see Cochard and Madariaga (1994) u(x,z) = [0,u(x,z),0] (8.2) The strain tensor has the following form e= 1 2   0 u,x 0 u,x 0 u,z 0 u,z 0   (8.3) The stress tensor σ=2µe+λtre reduces to σ=µ   0 u,x 0 u,x 0 u,z 0 u,z 0   (8.4) The equilibrium equation is expressed by µ∆u(x,z,t)= 0 in R3\Γc (8.5) Now u depends also on time t since the problem under consideration is quasistatic. Here ∆ denotes the Laplacian with respect to the variables x,z. The fundamental solution or the Green function for the last equation is, see Vladimirov (1984) G(x,z;ξ) = 1 2πµ lnr (8.6) where r2 =(x− ξ)2+z2. From Betti’s formula we find the displacement inside the elastic body u(x,z,t) =µ ℓ∫ −ℓ [[u(ξ,t)]] ∂G(x,z;ξ) ∂z dξ (8.7) The problem considered is two-dimensional, since it does not depend on y. 498 W.R.Bielski, J.J.Telega Now we have uN = 0 and the tangent displacement uT = u is in the direction of the axis y. We set [[u(x,t)]] =u(x,0+, t)−u(x,0−, t) (8.8) Then [[u(x,t)]] = 0 for |x|>ℓ [[u(x,t)]] 6=0 for |x| ¬ ℓ Eq. (8.5) is completed with (i) the sliding rule σ=σT ∈ ∂2D(ϑp, [[u̇]]) for |x|<ℓ p=1, . . . ,n (8.9) (ii) the evolution equation for ϑp ϑ̇p =Hp(ϑm, [[u̇]]) m,p=1, . . . ,n |x|<ℓ (8.10) (iii) the initial conditions ϑp(x,0)=ϑ 0 p(x) σ(x,0)=σ 0 (8.11) We observe that now the friction condition does not depend on the normal stress, cf Remark 8.1. In the case of the antiplane deformation, the dual problem (PG) reduces to: Find σ(x,t), x∈Γc, t∈ [0,T ] such that (8.9) is satisfied and ℓ∫ −ℓ ℓ∫ −ℓ [s(x)−σ(x,t)]G(x,z;ξ)σ̇(ξ,t) dξdx­ 0 ∀s(x)∈K1 (8.12) The indicator function of the set K1(ϑp) is a dual of D(ϑp, ·). Once the density of frictional dissipation is known, one can also find the set K1(ϑp). Eq. (8.2) is now trivially satisfied.We observe that in the case of the friction condition used in Cochard and Madariaga (1994), the problem (PG) for the antiplane shear does not involve Eq. (8.9). Remark 8.1. (i) The antiplane problem significantly simplifies the fault deformation. Here it has been used to show how the method of duality may be em- ployed to study the friction problem in the neighborhood of the fault. (ii) Okubo (1989) defined the fault as the plane x3 = 0 in an infinite, homogeneous, elastic whole space. In such case Γc = R 2 and our duality method can also be applied. Modelling contact problems... 499 9. Time and space discretizations Nowwepass to timediscretization of problem (PG) in the case of the anti- plane shear. First, we observe that thenGreen’s function G of the considered problemdoesnotdependon time t, cfEq. (8.6).Thuswehave dGσ/dt=Gσ̇. Therefore we can introduce the following approximation of the stress deriva- tive with respect to time t. Let the time interval [0,T ] be divided into L intervals (tl−1, tl) for l=1, . . . ,L and 0= t0 < t1 < ... < tL =T . The time derivative is approximated by the backward finite difference in the following way σ̇(x,tl)≈ σ(x,tl)−σ(x,tl−1) tl− tl−1 (9.1) The evolution equation of the internal variables takes the form ϑ̇p(x,t)=Hp(ϑm, [[u̇]]) After discretization in time we write ϑ̇p(x,tl)≈ ϑp(x,tl)−ϑ(x,tl−1) tl− tl−1 =Hp(ϑm(x,tl−1), [[u̇(x,tl−1)]]) (9.2) Substituting (9.1) into (PG) we get the following problem for the interval (tl−1, tl). After time discretization, the quasi-variational inequality (8.12) is written as the sequence of quasi-variational inequalities. Problem (P lG) Find σ(x,tl) and ϑp(x,tl), l = 1, . . . ,L; 0 = t0 < t1 < ... < tL = T , x∈ (−ℓ,ℓ) such that for all admissible stresses s= s(x)∈K1(ϑp(x,tl)), ℓ∫ −ℓ ℓ∫ −ℓ G(x,ξ)σ(ξ,tl)[s(x)−σ(x,tl)] dxdξ­ ­ ℓ∫ −ℓ ℓ∫ −ℓ G(x,ξ)σ(ξ,tl−1)[s(x)−σ(x,tl)] dxdξ (9.3) ϑp(x,tl)−ϑp(x,tl−1) tl− tl−1 =Hp(ϑm(x,tl−1), [[u̇(x,tl−1)]]) σ(x,0)=σ0(x) ϑ(x,0)=ϑ0(x) x∈ [−ℓ,ℓ] where G is given by (8.6) 500 W.R.Bielski, J.J.Telega Remark 9.1. 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Am., 88, 1466-1483 Modelowanie zagadnień kontaktowych z tarciem w mechanice uskoków geologicznych Streszczenie Cel pracy jest dwojaki. Po pierwsze, omówionomodele tarcia stosowane w geofi- zyce. Modele te obejmują tarcie zależne od strun, prędkości i poślizgu. Po drugie, zaproponowano nowy opis tarcia w języku nowoczesnej mechani- ki kontaktu, wprowadzając prawa poślizgu wiążące naprężenie poślizgu z prędko- ścią poślizgu. Prawa poślizgu sformułowano w postaci subróżniczkowej. Zagadnienia początkowo-brzegowe sformułowano w postaci silnej i wariacyjnej. Stosując funkcję Greena zaproponowano sformułowanie wariacyjne pozwalające wyznaczyć normalne i styczne naprężenia kontaktowe. Manuscript received March 20, 2001; accepted for print April 19, 2001