Nonsmooth dynamics of friction in modeling of earthquakes Vladimir Ryabov Future University-Hakodate, Japan
Earthquakes are caused by friction between tectonic plates From: http://seismo.berkeley.edu/
Stick-Slip on an Earthquake Fault Stick Slip
Stick-slip Reid s Hypothesis of Elastic Rebound Reid, H.F., The mechanics of the earthquake, v. 2 of The California earthquake of April 18, 1906. Report of the State Earthquake Investigation Commission, Carnegie Institution of Washington Publication 87, 1910. Earth, S. Marshak, W.W. Norton Elastic stress accumulates between the seismic events and is released during an earthquake. The elastic stress causes the earthquake in the sense that the elastic energy stored in the fault drives earthquake rupture.
Earthquake Models Two local mechanisms the stick-slip motion along a pre-existing fault; the brittle rock fracture Types of earthquake models (of the first kind): High-dimensional dynamical models: Multi-spring-block models (Burridge-Knopoff, Carlson- Langer) Constitutive friction law (Dieterich, Ruina, Scholz, ). Friction can depend on velocity, slip, prehistory, surface type. Cellular automata models (Nakanishi, Bak-Chen-Obukhov, Olami-Feder-Christensen, Sornette ) Low-dimensional dynamical models: One-block (Reid, Vasconcelos, Ryabov&Ito, ) Two-block (Huang&Turcotte, Ryabov&Ito, Galvanetto, de Sousa Vieira...), Three-block...
SINGLE DEGREE OF FREEDOM Equation of motion α r F r fr k el F r x 2 d x m 2 dt = kx F fr Simple velocity-weakening friction F fr F0 = 1+ γvr el F, γvr F0 1 1+ γv for v = 0 r r, for v r > 0
Andronov et al., <1937 Axle Break shoe
Velocity weakening friction: Comparison to an experiment From: J. Awrejcewicz and P. Olejnik, Int. J. Bifurcation and Chaos, Vol. 15, 1931 (2005).
J.H. Dieterich & A. L. Ruina constitutive rate and state dependent friction law α r F r fr k el F r x F fr dθ i dt N v = F* + Aln + θ i v* i= 1 = v L i θ i + B i ln v v * where L i are characteristic slip distances; - θ i state variables; B i friction parameters
Stick-Slip Instability Requires Some Form of Weakening: Velocity Weakening, Slip Weakening, Thermal/hydraulic Weakening, etc. Slip Weakening Friction Law Rate and State Dependent Friction Law µ s µ V o V 1 = e V o Slip rate µ d L Slip Stability Criterion µ d (v) a D c Slip b Stability Criterion Velocity Weakening b-a >0 K < K c ; Unstable, stick-slip K > K c ; Stable sliding K < K c ; Unstable, stick-slip K > K c ; Stable sliding
Can it be applied to earthquakes? Yes. For example in subduction zones Temperature Stiffness
Temperature (depth) dependence of friction From: Christopher H. Scholz, Earthquakes and friction laws, NATURE,391, 37(1998)
Problems with Dieterich-Ruina model Too many parameters Ambiguity in choosing parameters for controlling the given effect Not clear how many state variables in the friction law should be used Difficult to predict the behavior (analytically) and interpret the results in seismological terms
UNIFORM BURRIDGE-KNOPOFF MODEL (distributed system) 2 U t 2 = ξ 2 2 U s 2 U U φ 2αν + 2α t where U(s,t) is the relative displacement, v, the relative velocity of one side of a lateral fault motion, φ [.] nonlinear, slip and/or velocity-weakening friction law. Slip Weakening Friction Law Active fault Velocity Weakening Friction Law µ s L µ d µ d (v) Slip
One-dimensional chain of blocks and springsdiscrete Burridge-Knopoff model The equation of motion for the j-th block m d 2 dt X 2 j dx j ( ) X 2X + X k X F v + = kc j+ 1 j j 1 p j dt R.Burridge, L.Knopoff. Model and theoretical seismicity, Bull.Seismol.Soc.Am., 57, 3411-3471, 1967. J.M.Carlson, J.S.Langer. Mechanical model of an earthquake fault, Phys.Rev.A, 40, 6470-6484, 1989. S.L.Pepke, J.M.Carlson, B.E.Shaw. Prediction of large events on a dynamical model of a fault, J.Geophys.Res., 99, 6769-6788, 1994.
Guttenberg-Richter Law Per Bak, et. al, Phys. Rev. Lett. 2002 Despite of the apparent complexity of earthquakes, the probability distribution for their magnitudes follows a simple function known as the Gutenberg-Richter law: The number N (m) of earthquakes of magnitude M > m, N(m) = c 10 -bm, b 0.95 where the exponent b~1 is a universal exponent in the sense that it does not depend on particular geographic area. When plotted on a log-log scale, such power law appears as a straight line with the slope -b
Guttenberg-Richter law Model quality test Phys. Rev. A 1989 Phys. Rev. A 1991
Two-dimensional spring-block array 1-dim. 2-dim.
Cellular automation model (details of friction excluded) One-dimensional chain of blocks H. Nakanishi, Phys. Rev. A 43, 6613(1991) Olami-Feder-Christensen (OFC) model of earthquakes Z. Olami, H. J. S. Feder, and K. Christensen, Phys. Rev. Lett. 68, 1244 (1992).
Guttenberg-Richter (magnitude), Omori (aftershocks), and foreshocks Z. Olami, H. J. S. Feder, and K. Christensen, Phys. Rev. Lett. 68, 1244-1247 (1992) Contrary to other spring-block models, this one can be modified to demonstrate foreshocks and aftershocks S. Hergarten and H. J. Neugebauer, PRL, 2002
Problems in multi-block modeling Difficult to control (not clear what combination of parameters should be used, numerical experiment only) Huge computing resources for modeling Unpredictable behavior (oscillation regimes change with parameters in a complicated way)
What is the simplest model? α r F r fr k el F r x = periodic only one Degrees of freedom F fr dθ i = dt N v = F* + Aln + θ i v* i= 1 v L i θ i + B = i ln v v * α k p m = k c m k p x many Complex two degrees of freedom chaos is possible
Chaos in the two-block model Simple velocity-weakening friction F fr ( dx ) dt 1 1 1+γα α k p m k c m k p x α v 0 g dx dt F fr dx dt i v 0 1 = α + γ 1 γ 1 + 4 g 2 dxi dxi g α + at < v0 dt dt = 1 dxi at > v dxi dt 1 + γ α + dt 0 V. Ryabov and H. Ito (1995), Phys.Rev.E 52, No. 6, 6101-6112 M. de Sousa Vieira (1995), Phys.Lett. A 198, 407-414 U. Galvanetto and Sr. Bishop (1995) Chaos, Soliton Fract. 5, 2171-2179
Method of averaging Equations of motion dy dt dy dt dy dt dy dt 1 2 3 4 = y y 2 0 Form of solution 2 4 2 0 = ω y = = ω y 1 3 + κy + κy 3 1 G( y G( y 1 3 ) y ) y 2 4 CLOSE TO HARMONIC y 2 0.005-0.015-0.0115-0.0095 y 1 MULTIPLE SOLUTIONS
Results of averaging Hopf bifurcation condition Limit cycle amplitude α > α th 1 γ 1 4 γ + g 1 2 Limit cycle amplitude vanishes with α 0 Curves 1-5 correspond to increasing velocity α Saddle-node bifurcation Symmetry property x1 x 2 Complexity area (asymmetric attractors) is at small velocity values Symmetry breaking Curves 1-5 correspond to decreasing stiffness k
Typical road to chaos: symmetry breaking Block 1 Symmetric Block 2 y 2 0.025 y 4 0.025-0.015-0.011-0.007 y 1-0.015-0.011-0.007 y 3 Asymmetric y 2 0.025 y 4 0.025-0.015-0.011-0.007 y 1-0.015-0.011-0.007 y 3
Bifurcation diagram of the two-block system γ 60 40 20 AP-1 SN(T) H1 AP-1 SN(N) AP-1 PD ACH BD SP-1 H2 T BD SP-1 PD SCH BD SP-3 SN 0 0 0.01 0.02 0.03 α AP-1 SN(T) - saddle-node (theoretical) bifurcation leading to the arising of asymmetric attractors; AP-1 SN(N) - saddle node (numerical) bifurcation of asymmetric limit cycles arising in system (2); H1 - primary Hopf bifurcation; AP-1 PD - first period doubling of asymmetric periodic orbits; ACH BD - break down of the chaotic attractor originating from asymmetric solutions; SP-1 H2 - secondary Hopf bifurcation of a symmetric period-1 orbit, two-dimensional torus appears; T BD -torus break down; SP-1 PD - first period-doubling of the symmetric period-1 attractor; SCH BD - break down of the chaotic attractor originating from the symmetric orbit; SP-3 SN - period-3 saddle-node bifurcation. (from V. Ryabov and H. Ito (1995))
From continuous to discontinuous friction Chaos area shrinks as the friction becomes closer to the discontinuous function However, it remains finite Therefore, the case of discontinuous friction is less chaotic at g F fr dx ( ) dt 1 α v 0 g= dx dt
Conclusions from the analysis of the two-block system Dynamics is complex, as it can be expected for a system of two coupled nonlinear oscillators (see, e.g., Aronson, et al., Physica D 25 (1987) 20-104) Symmetry braking is a typical route to complex behavior, i.e. asymmetry is important for making a realistic earthquake model (see also: U. Galvanetto, Phys. Lett. A 293 (2002) 251) Method of averaging can be used successfully for predicting symmetry-breaking bifurcations in a broad range of control parameters Discontinuity of friction does not introduce drastic changes to the bifurcation pattern How useful are the results of bifurcation analysis for understanding the real earthquakes?
Is it really the simplest model? Three blocks? α r F r fr k el F r x = periodic only one Degrees of freedom F fr dθ i = dt N v = F* + Aln + θ i v* i= 1 v L i θ i + B = i ln v v * α k p m = k c m k p x many Complex two degrees of freedom chaos is possible
Fault motions: two-dimensional Normal fault Other names: normal-slip fault, tensional fault or gravity fault Reverse Fault Other names: thrust fault, reverse-slip fault or compressional fault Strike-Slip Fault Other names: transcurrent fault, lateral fault, tear fault or wrench fault Oblique-Slip Fault Combination of dip-slip faulting and strike-slip faulting From: http://www.iris.edu/gifs/animations/faults.htm
Back to one-block model but in two dimensions k x = k y equivalent to 1- dimensional case k x m F fr k y Equations of motion y x α Velocity-weakening friction r fr F0 F = r 1 + γ v where vr γ r is relative velocity is instability parameter Friction law d dt d 2 2 dt x 2 y 2 = k = k x y x + F y + F fr x fr y = k = k x y x γ r fr fr + F cos( ϕ ) r y + F fr sin fr ( ϕ ) F 0 0 tg(ϕ)=γ 0 1 2 3 4 5 Relative velocity
Direction of friction is as important as its absolute value fr el ϕ = ϕ + ϕ el v ϕ = ϕ ϕ r [ 1 exp( Ω v )] r F Stick el v=0 F fr velocity F el F v<<1 fr Start of slip Slip velocity v~1 F el F fr
Limit of α 0 in the 1-d model (G.L. Vasconcelos, Phys.Rev.Lett, 25, 4865 (1996) Friction law F 0 0 F fr tg(ϕ)=γ 0 1 2 3 4 5 Relative velocity Block displacement at v 0 Stickslip Stickslip Creep Creep velocity 1/Instability at v 0
STICK STATE BOUNDARY (SSB) 2 2 2 ( k x ) + ( k y ) x 0 y 0 = F0 k x > k y the block starts to move at SSB y F 0 k y F k 0 x x asymmetric model (x 0 ; y 0 ) α r ϕ 0 F el = ( k x k ) x 0 ; y y 0 restoring force=-shear stress
Stiffness k x,y and instability γ define dynamics k y ( γ ) F 2 0 4m INTERMITTENCY PERIODIC STICK- SLIP ONLY ( γ ) 0 0 4m STRONG ASYMMETRY OF ELASTIC FORCES: F 2 CREEP MOTION ONLY INTERMITTENTENCY k >> x k y k x
k x > k y Direction of motion α also x, α y defines the type of earthquakes α y k >> x k y ( ) α y intermittency creep intermittency periodic stick slip periodic stick slip periodic stick slip creep intermittency intermittency α x creep creep α x Critical point: α y α x cr = 4k x 2 2 0 γ F 2 0 F γ 4k 2 y
Intermittent motion k x k y y V. B. Ryabov & K. Ito, PAGEOPH, 158, 919 (2001). F fr x 0-0.1-0.2 Y -0.3-0.4-0.5. A B. creep α.c -0.6-0.03-0.01 0.01 X α large slip -0.184 V θ -0.188 A U X A' -0.192 A'' Y -0.196-0.2 α SSB -0.204-0.0244-0.024-0.0236 X
Creep motion From: Christopher H. Scholz, Earthquakes and friction laws, NATURE,391, 37(1998)
One-dimensional map (see also Galvanetto 1995) Intermittency Phase transition 5 4.5 4 Large slip event) θ n+1 3.5 3 2.5 laminar phase Creep motion 1.5 2.5 3.5 4.5 θ n Return to creep
Characteristic earthquakes 0.06 Occurrence probability 0.04 0.02 0 0 40 80 120 160 POISSON PROCESS Interevent time interval STICK-SLIP Occurrence probability 0,06 0,04 0,02 0 0 40 80 120 160 Interevent time interval [arbitrary units]
No correlation between successive events Creep motion is unstable (trajectories diverge in the slip phase) next big event is almost independent from the previous one 6000 4000 Not completely random??? T n+1 2000 0 0 2000 4000 6000 T n Unpredictable???
Understanding the dynamics of earthquakes 1. Anisotropy of stress plays a crucial role in the global type of seismicity pattern observed on a given fault segment (creep motion or earthquakes). Uniform elasticity produces periodic behavior. 2. Why do some faults creep while others stick and slip? This can be explained by different direction of tectonic plates motion with respect to the orientation of the stress vector on different faults or fault segments.
San-Andreas fault (from W. Kenneth Hamblin & Eric H. Christiansen. Earth s Dynamic Systems, Prentice Hall, New Jersey, 2001)
Fault geometry? A B
Understanding the dynamics of earthquakes Earthquakes α r x z Creep aseismic slip α rx z y y 3. Oblique subduction zones should demonstrate creep motion more often compared to the areas where the subducting plate moves normally to the continental crust.
Understanding the dynamics of earthquakes 4. Time intervals between successive large earthquakes is typically larger than those expected from elastic rebound theory (time-predictable model). STRESS Intermittency TIME Large stick-slip Twodimensional stick-slip Timepredictable model