Slip of a one-body dynamical spring-slider model in the presence of slip-weakening friction and viscosity ANNALS OF GEOPHYSICS, 59, 5, 2016, S0541; doi:10.4401/ag-7063 S0541 Slip of a one-body dynamical spring-slider model in the presence of slip-weakening friction and viscosity Jeen-Hwa Wang Institute of Earth Sciences, Academia Sinica, Taipei, Taiwan ABSTRACT This study is focused on analytic study at small displacements and nu- merical simulations of slip of a one-body dynamical slider-slider model in the presence of slip-weakening friction and viscosity. Analytic results with numerical computations show that the displacement of the slider is con- trolled by the decreasing rate, c, of friction force with slip and viscosity, h, of fault-zone material. The natural period of the system with slip- weakening friction and viscosity is longer than that of the system with- out the two factors. There is a solution regime for h and c to make the slider slip steadily without strong attenuation. The viscous effect is stronger than the frictional effect. Meanwhile, a change of h results in a larger effect on the slip of the slider than a change of c. Numerical sim- ulations are made for a one-body dynamical slider-slider model in the presence of three slip-weakening friction laws, i.e., the thermal-pres- surization (TP) friction law, the softening-hardening (SH) friction law, and a simple slip-weakening (SW) friction law, and viscosity. Results show that slip-weakening friction and viscosity remarkably affect slip of the slider. The TP and SW friction laws cause very similar results. The re- sults caused by the SH friction law are quite different from those by the other two. For the cases in study, the fixed points are not an attractor. 1. Introduction Essentially, the rupture processes of an earthquake consist of three steps: nucleation (or initiation), dy- namical propagation, and arrest. It is necessary to study the mechanisms controlling the whole rupture processes. Such processes are very complicated, and cannot be completely solved just using a simple model. Several factors, including brittle-ductile fracture rheol- ogy [Jeffreys 1942; Scholz 1990], normal stress [Fang et al. 2011], re-distribution of stresses after fracture, the geometry of faults [Fang et al. 2011], friction [Nur 1978; Dieterich 1979; Ruina 1983], distribution of frictional strengths [Wang and Hwang 2001; Wang 2008], heal- ing from dynamic to static friction after an earthquake [Wang 1997], pore fluid pressure [Scholz 1990; Wang 2009, 2011]; elastohydromechanic lubrication [Brodsky and Kanamori 2001; Garagash and Germanovich 2012]; thermal pressurization [Rice 2006; Bizzarri 2011a, 2011b], stress corrosion [Anderson and Grew 1977; Atkinson 1984], and metamorphic dehydration [Brantut et al. 2011], will control earthquake ruptures. Friction is one of the most important factors in controlling the rupture processes of an earthquake [Nur 1978; Dieterich 1979; Ruina 1983; Cao and Aki 1986; Knopoff et al. 1992; Rice 1993; Wang 1996, 1997, 2007, 2012; Rubin and Ampuero 2005; Ampuero and Rubin 2008; Bizzarri 2011c; Bhattacharya and Rubin 2014]. The friction coefficient, f, is defined as the ratio of shear stress, x, to the effective normal stress, veff , on the fault plane. Laboratory rock sliding experiments [Byerlee 1978] show that for most rock types, the friction coeffi- cient at which slip initiates is about 0.6 - 0.8. At lithostatic normal stress and hydrostatic pore pressure, observa- tions suggest that the shear strength of faults exceeds ~100 MPa at seismogenic depths on continental strike- slip faults. The frictional force between two contact planes is classically considered to drop from static one to dynamic one after the two planes move relatively. In- deed, the friction law that has been mainly inferred from laboratory experiments is quite complicated and not completely understood, especially for that on nat- ural faults due to a lack of observational constraints. This makes the proper constitutive law for fault friction an elusive mathematical formulation. Dieterich [1972] first found time-dependent static frictional strength of rocks in laboratory experiments. Dieterich [1979] and Shimamoto [1986] observed velocity-dependent fric- tional strengths. Dieterich [1979] and Ruina [1983] pro- posed empirical, velocity- and state-dependent friction laws. There are two state evolution laws, i.e., “aging” and “slip” versions. Essentially, velocity- and state-de- pendent friction includes two different processes: the velocity-weakening process and the velocity-hardening Article history Received May 27, 2016; accepted September 6, 2016. Subject classification: One-body spring-slider model, Displacement, Slip-weakening friction, Viscosity. one. In fact, a large debate related to the friction laws governing earthquake ruptures has been made for a long time. Although this problem is important, it is out of the scope of this article and thus will not be ex- plained in details. A detailed description of the gener- alized velocity- and state-dependent friction law and the debates can be found in several articles [Marone 1998; Wang 2002; Bizzarri and Cocco 2006c; Bizzari 2011c]. Several simple friction laws have been taken into account by some researchers. Burridge and Knopoff [1967] first considered a velocity-dependent, weakening- hardening friction law. Carlson and Langer [1989] pro- posed a purely nonlinearly velocity-weakening friction law which was also used by others [Carlson 1991; Carl- son et al. 1991; Beeler et al. 2008] to theoretically model earthquakes. Wang [1995, 1996, 2012] considered a piece- wise, linearly velocity-dependent weakening- harden- ing friction, which is simplified from the friction law proposed by Burridge and Knopoff [1967]. The de- creasing (weakening) and increasing (hardening) rates of dynamic friction with sliding velocity are two main pa- rameters of this friction law. Cao and Aki [1984/85] took a displacement softening-hardening friction law. Cochard and Madariaga [1994] and Madariaga and Cochard [1994] assumed that purely velocity-dependent friction models can lead to unphysical phenomena or mathematically ill-posed problems. This means that the velocity-dependent friction law is very unstable at low velocities both during the passage of the rupture front and during the possible slip arrest phase. Moreover, Ohnaka [2003] stressed that purely velocity-dependent friction is in contrast with laboratory evidence, that is, the friction law is not a one-valued function of velocity. Bizzarri [2011c] deeply discussed this point. Brune [1979] showed that frictional sliding at x≈10 MPa would produce a heat flow anomaly that is not ob- served in field data adjacent to the San Andreas fault [Lachenbruch 1980]. Temperature measurements in boreholes drilled across faults shortly after earthquakes also indicate low shear traction during seismic slip [Kano et al. 2006; Wang 2006, 2011]. Since Lachenbruch [1980] pointed out the heat-flux paradox, numerous studies have been placed on the thermal properties of the fault-zone rocks to address the importance of ther- mal effect during earthquake ruptures [Chester and Higgs 1992; Fialko 2004; Bizzarri and Cocco 2006a, 2006b; Wang 2006, 2007, 2009, 2011; Bizzarri 2011a, 2011b; Bizzarri and Crupi 2013]. Sibson [2003] also stressed the presence of pseudotachylytes [Sibson 1975] due to heating during earthquake ruptures. Numerous studies have conducted on modeling the frictional heat produced during seismic sliding. Theoretical studies on the spontaneous propagation of earthquake ruptures on 3D faults by Bizzarri and Cocco [2006a, 2006b] re- vealed that melting of rocks and fault gouge is likely to occur even with the inclusion of the thermal pressur- ization of pore fluids. Moreover, the dramatic fault weakening at high slip rates predicted by the flash heat- ing of micro-asperity contacts is not able to avert melt- ing [Bizzarri 2009]. When fluids are present in faults, thermal pressur- ization can play a significant role on earthquake rup- ture and also result in resistance on the fault plane [Sibson 1973; Fialko 2004; Bizzari and Cocco 2006a, 2006b; Rice 2006; Wang 2009, 2011, 2013; Bizzarri 2010; Bizzarri 2011a, 2011b]. Rice [2006] proposed two end- members models for thermal pressurization: the adia- batic-undrained-deformation (AUD) model and slip-on- a-plane (SOP) model. He also obtained the shear stress- slip functions caused by the two models. In spite of a large number of fault-governing fric- tion laws, the only constitutive law able to avoid the melting is a slip- and velocity-weakening friction law [Sone and Shimamoto 2009; Bizzarri 2010], for which the fault weakening is so dramatic that it cannot be counterbalanced by the resulting enhanced slip veloci- ties. However, both thermal pressurization of pore flu- ids and flash heating predict not only a very dramatic stress drop, but also a very high peak in fault slip veloc- ity, so that the final result is that melting temperature is very often exceeded, unless the slipping zone (where the deformation is concentrated) is extremely large [Biz- zarri and Cocco 2006b; Bizzarri 2009]. Bizzarri [2011a] stressed that when melting occurs, the rheological be- havior of the fault zone no longer obeys the Coulomb– Amonton–Mohr formulation, in that a viscous rheology is needed to describe the traction evolution during the ruptures. Jeffreys [1942] first emphasized the influence of viscosity on faulting. Viscosity can also be controlled by the presence of frictional melts in fault systems [By- erlee 1968]. Temperature, pressure, water content, etc., will influence viscosity [Turcotte and Schubert 1982]. Scholz [1990] suggested that the residual strength of fault-generated friction melts would be high and so present significant viscous resistance to shear. This in- hibits continued slip. On the other hand, Spray [1993, 1995] stated that most pseudotachylytes are partial melts possessing low viscosity, and capable of generating a sufficient melt volume to reduce the effective normal stress. Thus, friction melts can act as fault lubricants during co-seismic slip [Spray 2005]. His results show that viscosity remarkably decreases with increasing temperature. Rice et al. [2001] discussed the physical basis of velocity- and state-dependent friction, includ- ing the direct effect in thermally activated processes al- JEEN-HWA WANG 2 3 lowing creep slippage at asperity contacts on the fault surface. Wang [2007] stressed the viscous effect on earthquake ruptures from the comparison between simulated waveforms and seismograms recorded near the Chelungpu fault of the 1999 Chi-Chi, Taiwan, earthquake. Wang [2011] assumed that quartz plasticity could be formed in the main slip zone of the 1999 Chi- Chi, Taiwan, earthquake when T >300°C after the fault ruptured. The shear zone with quartz plasticity would be localized in a 5-mm thick heated layer. Quartz plas- ticity could lubricate the fault plane at higher T and yield viscous stresses to resist slip at lower T. On the other hand, several researchers [Knopoff et al. 1973; Cohen 1979; Xu and Knopoff 1994; Knopoff and Ni 2001; Dragoni and Santini 2015] took the viscous effect as a factor in causing seismic radiation to reduce energy during earthquake ruptures. Dragoni and Santini [2015] considered with two- degrees-of-freedom dynamical spring-slider model to approach two asperities on a fault. Except for the cou- pling between two sliders, the equation of motion of a slider in their model is essentially the same as that used in this study. Hence, their studies are explained in details more or less here. They introduced a term proportional to slip rate in the equations of motion to represent seismic radiation during the slipping modes. They gave a complete analytical solution of the four dynamic modes of the system. Any seismic event can be ex- pressed as a sequence of modes, for which the moment rate, the spectrum and the total seismic moment can be calculated. They also considered the energy budget of the event and calculated its seismic efficiency. Seis- mic radiation might change the evolution of the sys- tem from a given state, since it moves the boundaries between the different subsets of the sticking region. In addition, the slip amplitude in a seismic event is smaller, while the slip duration is longer in the presence of ra- diation (or viscosity). The shape of the moment rate function depends on the seismic efficiency and the seis- mic moment decreases with increasing efficiency at constant radiated energy (or constant viscosity). Since the ingredients of an ideal model are only partly understood, a set of equations to describe com- prehensively fault dynamics has not yet been estab- lished. Nevertheless, some models, for instance the crack model and dynamical lattice model, have been developed to approach fault dynamics for a long time. Although the frictional effect on earthquake ruptures has been widely studied as mentioned above, the stud- ies of viscous effect on earthquake ruptures are few. The viscous effect mentioned in Rice et al. [2001] was just an implicit factor included in the direct effect of friction law. In this work, I will explore the effects of slip-weakening friction, especially due to thermal pres- surization, and viscosity on earthquake ruptures on the basis of a one-body spring-slider model, which is used to approach an earthquake fault. The viscous effect is represented by an explicit parameter. Results will be ap- plied to understand earthquake ruptures. 2. Model One-body spring-slider model The equation of motion of the one-body dynam- ical spring-slider model (see Figure 1) is: md2u/dt2 = −K(u − vpt) −F (u,v) −U(v), (1) where m is the mass of the slider, u is the displacement of the slider, K is the spring constant, vp is the speed of the driving force, v = du/dt is the velocity of the slider, F is the frictional force between the slider and the back- ground and could be a function of u and v, and U is the viscous force between the slider and the background and is a function of v. The slider is pulled by a leaf spring of strength, K, with a constant velocity, vp, which represents the speed of a moving plate. When the driv- ing force, Kvpt is slightly larger than the static frictional force, Fo, friction changes from static friction strength to dynamic one. Viscosity For deformed materials, there are two compo- nents, i.e., elastic component and viscous component, when the viscous effect is present. The elastic compo- nent can be modeled as a spring with an elastic constant E, given by the formula: v=Ef, where v, E, and f are, respectively, the stress, the elastic modulus of the ma- terial, and the strain that occurs under the given stress. The viscous component can be modeled as a dashpot such that the stress–strain rate relationship can be given as, v=y(df/dt) where y is the viscosity of the material. There are two models to describe the viscous materials (cf. Hudson [1980]). The first one is the Maxwell model which can be represented by a purely viscous damper (denoted by “D”) and a purely elastic spring (denoted by “S”) connected in series, as shown in Figure 1. The model can be represented by the following equation: df/dt=dfD/dt+dfS/dt=v/y+E -1dv/dt. If the material is put under a constant strain, the stresses gradually relax. When a material is put under a constant stress, the strain has two components. First, an elastic com- ponent occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is applied. The Maxwell SLIP OF ONE-BODY SPRING-SLIDER MODEL model predicts that stress decays exponentially with time. One limitation of this model is that it does not predict creep accurately. The second one is the Kelvin-Voigt model, also known as the Voigt model, consists of a Newtonian damper and Hookean elastic spring connected in parallel, as shown in Figure 1. It is used to explain the creep be- havior of materials. The constitutive relation is expressed as: v(t) = Ef(t) + vdf(t)/dt, which represents a solid un- dergoing reversible, viscoelastic strain. Upon application of a constant stress, the material deforms at a decreasing rate, asymptotically approaching the steady-state strain. When the stress is released, the material gradually relaxes to its un-deformed state. At constant stress (creep), the model is quite realistic as it predicts strain to tend to v/E as time continues to infinity. The model is extremely good with modeling creep in materials, but with regards to relaxation the model is much less accurate. Although viscosity varies with temperature, pres- sure, and water content, only a constant viscosity for each segment is considered below. The Newtonian vis- cous force is described by a dash-pot shown in Figure 2 specified with viscosity y between the slider and the moving plate, and, thus, the viscous force at the slider is represented by −yv where v is the velocity of the slider. For the Kelvin-Voigt model, the stress is a func- tion of both strain and strain rate and thus can be ap- plied to the seismological problems [Hudson 1980]. However, it is not easy to directly implement vis- cosity in a dynamical system as used in this study. Hence, viscosity is here represented in an alternative way. Viscosity leads to the damping of oscillations of a body. The damping coefficient is usually proportional to viscosity and is controlled by the linear dimension of the body in a viscous fluid. For example, according to Stokes’ law, the damping coefficient h of a sphere of radius R in a fluid of viscosity o is given by h= 6rRy (cf. Kittel et al. [1968]). In order to simplify the problem, the damping coefficient is regarded as viscosity here- after. Hence, the viscous force is U=hv. Noted that the unit of h is N/(m/s). Friction due to thermal pressurization Equation (1) exhibits that the motion of the slider is controlled by the frictional force, F (u,v). As men- tioned above, friction can also be produced from ther- mal pressurization. On a fault plane with an area of A and an average displacement u–, the frictional energy caused by the dynamic friction stress, xd, is Ef = xdu –A which could result in a temperature rise, DT. Frictional heat can conduct outwards from the slipping zone to wall rocks. Theoretical analyses [Fialko 2004; Bizzari and Cocco 2006a, 2006b] show that T is described by an error function of distance and decays outwards from the fault plane. Under thermal pressurization, the en- ergy and fluid mass conservation equations in a 1-D fault plane, in which the x- and y-axes denote the di- rections along and normal to the fault plane, respec- tively, can be found in Rice [2006]. Rice [2006] proposed two end-members models for thermal pressurization: the adiabatic-undrained- deformation (AUD) model and slip-on-a-plane (SOP) model. The first model corresponds to a homoge- neous simple shear strain at a constant normal stress vn on a spatial scale of the sheared layer that is broad enough to effectively preclude heat or fluid transfer. The second model shows that all sliding is on the plane with x(0) = f (vn− po) where po is the pore fluid pressure on the sliding plane (y = 0). For this second model, heat is transferred outwards from the fault plane. The shear stress- slip functions, x(u), caused by thermal pressurization [Rice 2006] are: (a) xaud(u) = f (vn− po)exp(−u/uc) for the AUD model; and (b) xsop(u) = f (vn− po) exp (u/L*) erfc (u/L*) 1/2 for the SOP model, The two parameters uc and L* are the respec- tive characteristic displacements, which are in terms of physical properties of the fault-zone materials, to con- trol the shear stress-slip functions of the individual end- member models. For the AUD model, uc is associated with the thickness of fault zone and a characteristic dis- placement of the shear stress-slip function. For the SOP model, Rice [2006] addressed that there is not a char- JEEN-HWA WANG 4 Figure 2. The two types of viscous materials: (a) for the Kelvin– Voigt model and (b) for the Maxwell model. (l= spring constant and y= coefficient of viscosity). Figure 1. One-body spring-slider model. In the figure, u, K, h, vp, N, and F denote, respectively, the displacement, the spring constant, the coefficient of viscosity, the velocity of the driving force, the nor- mal force, and the frictional force. 5 acteristic displacement for the shear stress-slip function. The stress xaud(u) displays exponentially slip-weakening friction. Indeed, The stress xsop(u) also shows slip-weak- ening [Wang 2009]. Since the SOP model is based on a constant velocity, it is not appropriate in this study. In order to perform the analytical study of the fric- tional effect on earthquake ruptures based on the one- body spring-slider model, a simplified linearly slip- weakening friction law in the form of F (u) = Fo−cu [Wang 2016], where c is the weakening rate, is consid- ered hereafter. This leads to the marginal analyses of slip of one-body spring-slider model in the presence of friction, and the results could work for slow earth- quakes. For numerical simulations of the dynamical model, a slip-weakening friction law (denoted by the TP friction law hereafter): F (u) = Fo exp(−u/uc), where uc is a characteristic displacement, from the stress-slip function of the AUD model, is first taken into account. In addition, the displacement softening-hardening fric- tion law (denoted by the SH friction law hereafter): F (u) = Fo exp[−(u 2−uc 2 )/c2 ], where Fo , uo, and c are con- stants, used by Cao and Aki [1984/85] is considered. For the purpose of comparison, a slip-weakening friction law (denoted by the SW friction law hereafter): F (u) = Fo/(1+u/uc) similar to the velocity-weakening friction law: F (u) = Fo/(1+v/vc), where Fo is the static frictional force and vc is the characteristic velocity, proposed by Carlson and Langer [1989] is also used. The functions of normalized friction force versus displacement for the three slip-weakening friction laws, with uc= 0.1 m and c = 0.3 m, are displayed in Figure 3. Obviously, the SH friction law that increases slightly with slip and then de- creases with increasing slip is quite different from the other two. For the given values of uc and c, the value of SH friction is higher than those of SW and TP friction when the displacement is shorter than a certain value. The value of SW friction is higher than that that of TP friction and the difference between them increases with displacement. Obviously, the TP friction law leads to a faster drop of friction and a higher stress drop than the other two. 3. Analytical studies Substituting the simplified slip-weakening friction law, i.e., F (u) = Fo−cu, and viscous force, i.e., U(v) =hv, into Equation (1) leads to md2u/dt2 = −K(u−vpt) − Fo+ cu − hv. (2) When the driving force, Kvpt is slightly larger than Fo, the frictional force changes from static one to dy- namic one and thus the slider moves. Since the dura- tion time of an earthquake rupture is usually short and vp (≈10 −11 m/s) is also very small, the value of Kvpt dur- ing an earthquake can be ignored. Hence, the equation of motion becomes: md2u/dt2 +hdu/dt + (K−c)u = 0. (3) Inserting the trail solution ei t into Equation (3) leads to ma2 − iha− (K−c) = 0. (4) The solutions of Equation (4) are a= (ih/2m) ± [K/m − (h2 + 4mc)/4m2]1/2. (5) Inserting Equation (5) into eiat leads to e−ht/2mexp {±i [(K/m) − (h2 + 4mc)/4m2]1/2 t}. The first term shows attenuation of slip with time and the second one rep- resents the slip of the slider. The slip remarkably de- creases when viscosity is high or the mass of the slider is small. Let ~o=(K/m) 1/2 be the natural angular fre- quency of the one-body spring-slider system without friction and viscosity. The natural period is To= 2r/~o= 2r(m/K)1/2. The natural angular frequency of the pres- ent system with friction and viscosity is ~n= [~o 2 − (h2 + 4mc)/4m2]1/2. (6) Equation (6) indicates that ~n is an imaginary num- ber when ~o<(h 2 + 4mc)1/2/2m. Together with the term e−ht/2m, this inequality results in thorough attenuation of slip. Hence, the condition of ~o>(h 2 + 4mc)1/2/2m is necessary for the existence of stable slippage of the slider. Under this condition, ~n is lower than ~o. In other words, the natural period of the system, i.e., Tn= SLIP OF ONE-BODY SPRING-SLIDER MODEL Figure 3. Three slip-weakening friction laws: the solid line for F(u) = 1/(1+u/uc) with uc= 0.1 m; the dashed line for F(u) = exp[−(u 2−uc 2)/c2] with uc= 0.1 m and c = 0.3 m; and the dotted line for F(u) = 1/(1+u/uc) with uc= 0.1 m. 2r/~n, is Tn= To/[1 − To 2 (h2 + 4mc)/(4rm)2]1/2. (7) Obviously, Tn is longer than To and increases with h and c, thus indicating that friction and viscosity both lengthen the natural period of the system. In order to further describe the condition: ~o> (h2 + 4mc)1/2/2m, this inequality is re-written as mK > (h/2)2 + mc. Figure 4 displays the curve of parabolic equation: h2 + 4mc= 4mK. This gives c= K when h= 0 and h2= 2(mK)1/2 when c= 0. Since the four parameters are all positive, the values of h and c that lead to a real value of ~n for stable slippage of the slider must be inside the range (called the solution regime) below the curve. Let the general solution of Equation (3) be u(t) = C1exp(ia1t) + C2exp(ia2t) with a1= ih/2m +~n and a2= ih/2m −~n. This gives u(t) = e −ht/2m [C1exp(i~nt) + C2exp (−i~nt)]. The velocity of the slider is v(t) = (−h/2m) e−ht/2m[C1exp(i~nt) + C2exp(−i~nt)]+i~ne −ht/2m ×[C1exp (i~nt) + C2exp(−i~nt)]. In order to solve the problem, the initial conditions are: (1) the displacement of the slider is null at t = 0, i.e., u(0) = 0; and (2) the initial ve- locity is vo at t = 0, i.e., v(0) = vo. The lower bound of vo should be vp because the plate motion is always in op- eration. This gives C1+ C2= 0; (8a) (−h/2m + i~n)C1− (h/2m + i~n)C2= vo. (8b) The solutions of Equation (8) are C1= vo/2i~n and C2= −vo/2i~n. Inserting the solutions with C1 and C2 into Equation (3) leads to u(t) = e−ht/2m (vo/~n)[exp(i~nt) −exp(−i~nt)]/2i. This gives u(t) = (vo/~n) e −ht/2m sin (~nt), (9) where the ratio vo/~n denotes the amplitude of slip function and varies with four model parameters. The value of u(t) increases with time, reaches the peak value and then decreases with time. The peak value of u(t) with v(t) = 0 occurs at time tp = tan −1(2m~n/h). In order to schematically demonstrate Equation (9) in terms of c and h, a few computational examples for the normalized displacement, i.e., u(t)/(vo/~n), with K = 10 nt/m and m = 10 kg, which lead to ~o= 1 Hz or To= 2r sec, are given below. Based on Figure 3, c and h must be smaller than 10 nt/m and 20 nt/(m/s), re- spectively, when K = 10 nt/m and m = 10 kg. Figure 5 exhibits u(t)/(vo/~n) for five values of c, i.e., 0, 2, 4, 6, and 8 nt/m, when h= 0 nt/(m/s). Since the displacement decays very fast with h, only u(t)/(vo/~n) for five small values of h, i.e., 0.0, 0.2, 0.4, 0.6, and 0.8 nt/(m/s), when c= 1 nt/m are displayed in Figure 6. In Figures 5 and 6, JEEN-HWA WANG 6 Figure 4. The solution regime of h and c based on K described by the parabolic equation: h2 + 4mc= 4mK for the one-body spring- slider model. Figure 5. The slip of the slider for five values of c, i.e., 0, 2, 4, 6, 8 N/m (from left to right), when K = 10 N/m, m = 10 kg, and h = 0. Figure 6. The slip of the slider for five values of h, i.e., 0.0, 0.2, 0.4, 0.6, 0.8 N/(m/s) (upside down), when K = 10 N/m, m = 10 kg, and c= 1 N/m. 7 the normalized displacements for t > tp are displayed by dashed lines, because the slider stops when v(tp) = 0. It is noted that since ~n decreases with h and c, the actual amplitude should increase with both h and c. 4. Numerical simulations In numerical simulations, above-mentioned non- linear friction laws with linear viscous law are taken into account. Substituting the TP friction law and the vis- cous law into Equation (3) leads to md2u/dt2 = −K(u−vpt) − Foexp(−u/uc) − hv. (10) In order to easily perform numerical computations, Equation (10) must be normalized. The normalization parameters are: Do = Fo/K, U = u/Do, Uc= uc/Do~o, and Vp= vp/Do~o and x=(K/m) 1/2 t = ~ot. This gives dx= ~odt, du/dt = [Fo/(mK) 1/2] dU/dx, d2u/dt2 = (Fo/mK) d2U/dx2, A = d2U/dx2, and V = dU/dx. In addition, h/(mK)1/2 is simply denoted by h below. Clearly, all nor- malized parameters are dimensionless. Hence, Equa- tion (10) becomes: d2U/dx2 = −U − hdU/dx− exp(−U/Uc) +Vpx. (11) Let y1= U and y2= dU/dx. Equation (11) can be re- written as two first-order differential equations: dy1/dx= y2 (12a) dy2/dx= −y1− hy2− exp(−y1/Uc) + Vpx. (12b) For the SH friction law, Equation (11) can be nor- malized and re-written as two first-order differential equations with the normalized parameters: dy1/dx= y2 (13a) dy2/dx= −y1− hy2− exp[−(y1 2 −Uc 2)/|2] +Vpx. (13b) where |= c/Do. For the SW friction law, Equation (11) can be nor- malized and re-written as two first-order differential equations with the normalized parameters: SLIP OF ONE-BODY SPRING-SLIDER MODEL Figure 7. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized cumulative displacement ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax of the slider, acted by TP friction of F(U) = exp(−U/Uc), for h= 10 (solid line), 20 (dashed line), and 30 (dotted line) when Uc = 1. dy1/dx= y2 (14a) dy2/dx= −y1− hy2− 1/(y1+ Uc) +Vpx. (14b) Equations (12) - (14) will be numerically solved using the fourth-order Runge-Kutta method (cf. Press et al. [1986]). It is noted that only the positive displace- ment is considered in this study, because the negative values of slip are usually not admitted in seismological applications. For each friction law, four diagrams are pro- duced from numerical simulations: the time variations in normalized acceleration, A/Amax, the time variations in normalized velocity, V/Vmax, the time variations in nor- malized cumulative displacement, ∑U/(∑U)max, where ∑U denotes the cumulative displacement, and the phase portrait of V/Vmax versus U/Umax. A phase portrait, denoted by y = f (x), is a plot of a physical quantity versus another of an object in a dy- namical system [Thompson and Stewart 1986]. The in- tersection point of the bisection line, i.e., y = x, and f (x) is called the fixed point, that is, f (x) = x. If the function f (x) is continuously differentiable in an open domain near a fixed point xf and |f ’(xf )|<1, attraction is gen- erated. In other words, an attractive fixed point is a fixed point xf of a function f (x) such that for any value of x in the domain that is close enough to xf, the iter- ated function sequences, i.e., x, f (x), f 2(x), f 3(x),…, con- verges to xf . An attractive fixed point is a special case of a wider mathematical concept of attractors. Chaos can be generated at some attractors. The details can be seen in Thompson and Stewart [1986] or other nonlin- ear literatures. Figure 7 displays the time variations in A/Amax, V/Vmax, and ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax with h=10 (solid line), 20 (dashed line), and 30 (dotted line) when Uc= 0.1 for the TP friction law. This figure shows the viscous effect. Figure 8 dis- plays the time variations in A/Amax, V/Vmax, and ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax for the TP friction law with Uc= 0.1 (solid line), 0.3 (dashed line), and 0.5 (dotted line) when h=10. Fig- ure 9 displays the time variations in A/Amax, V/Vmax, and ∑U/(∑U)max and the phase portrait of V/Vmax ver- sus U/Umax for the SW friction law with Uc= 0.1 (solid JEEN-HWA WANG 8 Figure 8. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized cumulative displacement ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax of the slider, acted by TP friction of F(U) = exp(−U/Uc), for Uc = 0.1 (solid line), 0.3 (dashed line), and 0.5 (dotted line) when h= 10. 9 line), 0.3 (dashed line), and 0.5 (dotted line) when h=10. Figure 10 displays the time variations in A/Amax, V/Vmax, and ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax for the SH friction law with Uc= 0.1 (solid line), 0.3 (dashed line), and 0.5 (dotted line) when |= 0.3 and h=10. Figures 8-10 exhibit the frictional effect. 5. Discussion Analytic results at low velocities leads to Equation (9) which reveals the influences on slip of the slider by friction and viscosity. When c= 0, Equation (9) demon- strates the slip of the slider specified by an exponen- tially damped sinusoidal function. Friction only lengthens the natural period of the system, Tn; while viscosity not only lengthens Tn but also makes the wave attenuated. When h= 0, Equation (9) expresses a purely sinusoidal wave. Figure 5 shows that the time functions of displace- ments exhibit a sinusoidal function because of h= 0 and change with c. Since the normalized displacement and h= 0 are considered, the peak amplitude does not change with c. Actually, the displacement itself de- pends upon h and ~n which are a function of c, h, K, and m. The occurrence time of peak amplitude and the predominant period increase with c. The initial chang- ing rates of displacement with time are almost the same for the five values of c. Results suggest that slip- weakening friction can influence slip of the slider. Figure 6 exhibits that the time functions of dis- placements show a sinusoidal function (displayed by the upmost curve) when h= 0, and departs from a si- nusoidal function when h> 0. The amplitude decreases with increasing h. Meanwhile, the peak amplitude ap- pears earlier when h becomes larger. This might be due to a fact that attenuation of slip wave remarkably in- creases with h. Based on Equation (7), the predominant period should increase with h. This point cannot be clearly viewed from Figure 6 due to strong attenuation. Unlike Figure 5, the initial changing rate of displace- ment with time slightly decreases with increasing h. Re- sults suggest that viscosity plays a significant role on slip of the slider. A comparison between Figure 5 and Figure 6 reveals that a change of h will result in a larger SLIP OF ONE-BODY SPRING-SLIDER MODEL Figure 9. The time variations in normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized cummulative displacement ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax of the slider, acted by SW friction of F(U) = 1/(U + Uc), for Uc= 0.1 (solid line), 0.3 (dashed line), and 0.5 (dotted line) when h= 10. effect on slip of the slider than a change of c. The viscous effect on slip of the slider in the pres- ence of TW friction is displayed in Figure 7 which shows the time functions of A/Amax, V/Vmax, and ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax of the slider for h=10 (solid line), 20 (dashed line), and 30 (dotted line) when Uc= 0.1. Figure 7a shows that the amplitude of A/Amax does not change with h, but the occurrence time of the amplitude in- creases with h. The predominant period of the time function of A/Amax increases with . Figure 7b shows that the amplitude of V/Vmax does not change with h, but the occurrence times of the amplitude increase with h. The predominant period of the time function of V/Vmax increases with Uc. Figure 7c shows that the value of ∑U/(∑U)max decreases with increasing h. Fig- ure 7d shows the phase portrait of V/Vmax versus U/Umax. The three portraits are almost coincided. In the plot, the intersection points of the bisection line (denoted by a thin solid line) with the three curves are the fixed points. Although the slope is not calculated for each fixed point, the value can be estimated from the plot. The absolute values of slope at the fixed points are almost the same and are likely all larger than 1, and thus the fixed points are not an attractor. The frictional effects caused by three different slip- weakening friction laws in consideration are displayed in Figures 8-10. Figure 8 shows the time functions in A/Amax, V/Vmax, and ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax of the slider, acted by TW friction, for Uc= 0.1 (solid line), 0.3 (dashed line), and 0.5 (dotted line) when h=10. Figure 8a shows that the amplitude of A/Amax obviously change with Uc, and the occurrence time of the amplitude increases with Uc. The predominant period of the time function of A/Amax increases with Uc. Figure 8b shows that the am- plitude of V/Vmax does not change with Uc, but the oc- currence time of the amplitude increases with Uc. The predominant period of the time function of V/Vmax in- creases with Uc. Figure 8c shows that the value of ∑U/(∑U)max decreases with increasing Uc. Figure 8d shows the phase portrait of V/Vmax versus U/Umax. In the plot, the intersection point of the bisection line (de- noted by a thin solid line) with each of the three curves JEEN-HWA WANG 10 Figure 10. The time variations in normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized cumulative displace- ment ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax of the slider, acted by SW friction of F(U) = exp[−(U 2−Uc 2)/|2], for Uc= 0.1 (solid line), 0.3 (dashed line), and 0.5 (dotted line) when |= 0.3 and h= 10. 11 is the fixed point. Although the slope is not calculated for each fixed point, the value can be estimated from the plot. The absolute values of slope at the fixed points increases with Uc and are all larger than 1. Hence, the fixed points are not an attractor. Figure 9 shows the time functions in A/Amax, V/Vmax, and ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax of the slider, acted by SW fric- tion, for Uc= 0.1 (solid line), 0.3 (dashed line), and 0.5 (dotted line) when h=10. Figure 9a shows that the am- plitude of A/Amax changes with Uc, and the occurrence time of the amplitude increases with Uc. The predom- inant period of the time function of A/Amax increases with Uc. Figure 9b shows that the amplitude of V/Vmax does not change with Uc, and the occurrence time of the amplitude increases with Uc. The predominant pe- riod of the time function of V/Vmax increases with Uc. Figure 9c shows that the value of ∑U/(∑U)max decreases with increasing Uc. Figure 9d shows the phase portrait of V/Vmax versus U/Umax. In the plot, the intersection point of the bisection line (denoted by a thin solid line) with each of the three curves is the fixed point. Al- though the slope is not calculated for each fixed point, the value can be estimated from the plot. The absolute values of slope at the fixed points increases with Uc and are all larger than 1. Hence, the fixed points are not an attractor. A comparison between Figure 8 and Figure 9 suggests that the TP and SW friction laws likely make similar effects on earthquake ruptures. This might be due to similar variations in friction force with slip for the two friction laws as displayed in Figure 3. Figure 10 shows the time functions in A/Amax, V/Vmax, and ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax of the slider, acted by SW fric- tion, for Uc= 0.1 (solid line), 0.3 (dashed line), and 0.5 (dotted line) when h=10. Unlike Figures 7-9, Figure 10a shows that A/Amax drops suddenly from an initial value to zero and almost does not change with Uc. Fig- ure 10b shows that the time functions of V/Vmax and the occurrence times of the peak value of V/Vmax do not change with Uc. Figure 10c shows that the time functions of ∑U/(∑U)max only slightly increases with Uc. Figure 10d shows that the phase portraits of V/Vmax versus U/Umax for the three values of Uc are al- most the same and form a line intersecting the bisec- tion line. at the respective fixed points. The three fixed points are almost the same. Although the slope is not calculated for each fixed point, the value can be esti- mated from the plot. The absolute values of slope at the fixed points are likely all slightly larger than 1, and thus the fixed points are not an attractor. In addition, the plots in Figure 10 are quite different from those in Figures 8-9. This might be due to a fact that the SW friction law is quite different from the other two as dis- played in Figure 3. 6. Conclusions In order to study the effects on earthquake rup- tures caused by slip-weakening friction and viscosity, the slip of a one-body dynamical slider-slider model is theoretically analyzed and numerically simulated when the two factors are present. Analytic results with nu- merical computations show that the displacement of the slider is controlled by the decreasing rate, c, of fric- tion force with slip and viscosity, h, of fault-zone material. The natural period of the system with slip-weakening friction and viscosity is longer than that of the system without the two factors. There is a solution regime controlled by the parabolic equation of h2 + 4mc= 4mK to result in stable slippage of the slider. Meanwhile, a change of h will result in a larger effect on slip of the slider than a change of c. Numerical simulations lead to the time functions in A/Amax, V/Vmax, and ∑U/(∑U)max and the phase portrait of V/Vmax versus U/Umax of the slider in the presence of three slip-weakening friction laws, i.e., the thermal- pressurization (TP) friction law, the softening-harden- ing (SH) friction law, and a simple slip-weakening (SW) friction law, and viscosity. Results show that slip-weak- ening friction and viscosity remarkably affect the three time functions and phase portrait. The TP and SW fric- tion laws cause very similar results. The results caused by the SH friction laws are quite different from those by the other two. 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