1 / 49

Dynamical Spring-slider (Lattice) Models for Earthquake Faults

Dynamical Spring-slider (Lattice) Models for Earthquake Faults. Jeen-Hwa Wang, Institute of Earth Sciences, Academia Sinica. Earthquake Fault and Seismic Waves (An Example of the Chelungpu Fault along which the 1999 Chi-Chi Earthquake happened). Viewpoints about a Fault Zone.

anais
Download Presentation

Dynamical Spring-slider (Lattice) Models for Earthquake Faults

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Dynamical Spring-slider (Lattice) Models for Earthquake Faults Jeen-Hwa Wang, Institute of Earth Sciences, Academia Sinica

  2. Earthquake Fault and Seismic Waves(An Example of the Chelungpu Fault along which the 1999 Chi-Chi Earthquake happened)

  3. Viewpoints about a Fault Zone • Geologists: A narrow zone with complex cataclastic deformations • Rock Scientists: A narrow zone with gouge and localized deformations • Seismologists: One or several double couples of forces exerting on a well-defined ruptured plane • Physicists: A domain of first-order phase transition • Mathematicians: ? (I do not know.)

  4. Ingredients and Capability of Models Simulating Earthquake Faults A Minimal Set of Ingredients: Current Capability: 1. Model: modest (e.g. spring-slider model and crack model ) 2. Constitution law of friction: incomplete 3. Initial condition: unknown • 1. Plate tectonics: to restore energy dissipated in faulting and creeping • 2. Ductile-brittle fracture rheology • 3. Stress re-distribution after fractures • 4. Thermal and fluid effects • 5. Healing process • 6. Non-uniform fault geometry

  5. Models for Earthquake Faults A Comprehensive Set of Models: Basic Models: Crack model (Griffith, 1922) (the most commonlly used model) One- to many-body dynamical spring-slider (lattice) models (Burridge and Knopoff, 1967) Crustal-scaled model(Sornette and Sornette, 1989) Granular mechanics model(Moral & Place, 1993) 1. Statistical Model: Verse-Jones (1966) 2. Stochastic Model (Knopoff, 1971) 3. Stochastic/Physical Model: (a) M8 Algorithm (Keilis-Borok et al., 1988) (b) Pattern Dynamics (Rundle et al., 2000) 4. Physical Model: A. Crack Models: (a) Quasi-static Model (Stuart, 1986) (c) Quasi-dynamic Model (Mikumo & Miyatake, 1978) (d) Crack Fusion (Newman & Knopoff, 1982) B. Dynamic Models: (a) Spring-slider (Lattice) Model (Burridge & Knopoff, 1967) (b) Block Model (Gabrielov et al., 1986 ) (e) Crustal-scaled Model (Sornette and Sornette, 1989) (d) Granular Mechanics Model (Moral & Place, 1993) C. Statistical Physics Models: (a) SOC Model (Bak & Tang, 1989) (b) Percolation Model (Otsuka, 1972) (c) Fluctuation Model (Rundle & Kanamori, 1987) (d) Renormalization Model (Katz, 1986; Turcotte, 1986) (e) Fractal Model (Andrews, 1980) (f) Growth Model (Sornette, 1990) (g) Traveling Density Wave Model (Rundle et al., 1996)

  6. 1-D N-body Spring-slider Model The equation of motion at the i-th slider (Burridge and Knopoff, BSSA, 1967): m(d2ui/dt2)=Kc(ui+1-2ui+ui-1)-Kl(ui-Vpt)-F(qi,vi) where ui=the slip of the i-th slider, measured from its initial equilibrium position (m) vi(=dui/dt)=the velocity of the i-th slider (m/s) m=the mass of a slider (kg) Vp=the plate moving speed (m/s) Kc=the strength of a coil spring (coupling between two sliders) (nt/m) Kl=the strength of a leaf spring (coupling between the plate and a slider) (nt/m) F(qi,vi)=a velocity- and state-dependent friction force (nt) qi=the state parameter of the i-th slider.

  7. Classical Friction Law Fo F (friction force) Fd v Velocity Fo: the static frictional force (Breaking strength) Fd: the dynamic frictional force

  8. (a-b)>0: strengthening or hardening (a-b)<0: weakening or softening Velocity- and State-dependent Friction Law Direct Effect Evolution Effect The factor a-b is a function of sliding velocity, temperature, loading rate etc. 5 cm/year

  9. Commonly-used Velocity- and State-dependent Friction Law One-state-variable Velocity- and State- dependent Friction Law: m=mo+a(v/vo)+bln(voq/d) The laws describing the state variable, q: Slowness law: dq/dt=-(vq/d)ln(vq/d) Slip law: dq/dt=-(vq/d)

  10. Shear Stress (or Friction) versus Slip due to Thermopressurization (Wang, BSSA, 2011)

  11. Simplified Velocity-weakening Friction Law Fo F (friction force) rw rh gFo vc Velocity Fo: the static frictional force gFo: the minimum dynamic frictional force (0<g<1) vc: the characteristic velocity with F=gFo rw: the decreasing rate of friction force with velocity rh: the increasing rate of friction force with velocity (healing of friction)

  12. Boundary Conditions • Periodic BC: u1=uN • Fixed BC: u1=uN=0 • (Stress-) Free BC: du1/dx=duN/dx=0 • Absorption BC: several ways • Mixed BC

  13. Main Model Parameters 1. s=KL/KC: stiffness ratio (coupling factor) s>1: weakly coupling between the plate and the fault s<1: strongly coupling between the plate and the fault 2. rw=the decreasing rate of friction force with velocity rh=the increasing rate of friction force with velocity 3. g: the friction force drop factor (0<g<1) 4. Vp: the plate velocity (10-9 m/sec) 5. D: fractal dimension of the distribution of the breaking strengths (or static friction), Fs 6. R: roughness of fault strengths [=(Fsmax-Fsmin)/Fsmean] 7. m: the mass of a slider ( inertial effect)

  14. Some Properties of the Spring-slider Model 1. There is no characteristic length. (=> a good model for SOC) 2. The system becomes unstable when a small perturbation is introduced. (Two ways to arrest a rupture: a. inhomogeneous frictional strength; b. velocity-weakening-hardening friction force.) 3. Intrinsic complexity a. Nonlinear friction (Carlson and Langer, 1989) b. Heterogeneous frictional strengths (Rice, 1993) h: the size of a nucleation size h*=2mdc/p(b-a)max Lc: the characteristic size h>h*=> chaotic behavior h<h* => periodic behavior For the spring-slider models, h*=0 => chaotic behavior 4. Nearest-neighbors effect (=> Short-range effect) 5. Two time scales: a. inter-event time (several hundred or thousand years) b. rupture duration time (several ten seconds)

  15. Three Rupture Modes in the 1-D Model(Wang, BSSA, 1996) C2=Co2+[Kl/m-(rw/2m)2]/k2 C: the propagation velocity of motions of sliders Co: the propagation velocity of motions of sliders in the absence of both Kl-spring and friction (This is the P-wave velocity.) (1) rw<2(mKl)1/2 => C>Co (Supersonic ruptures) (2) rw=2(mKl)1/2 => C=Co (Sonic ruptures) (3) rw>2(mKl)1/2 => C<Co (Subsonic ruptures)

  16. S=50 rw=1 g=0.8 S=100 rw=1 g=0.8 S=50 rw>>1 g=0.6 S=100 rw>>1 g=0.6 D=1.5; R=0.5 Wang (1995)

  17. Need a Two-dimensional Model • A 2-D dynamic model, with a more realistic constitution law of friction, is strongly needed for the studies of earthquakes and seismicity. Ma et al. (2003)

  18. 2-D N×M-body Dynamical Model(Wang, BSSA, 2000, 2012) The equations of motion of the (i, j) slider are: m2ujk/t2=K[u(j+1)k-2ujk+u(j-1)k]+eK[uj(k+1)-2ujk+u]+eK[(w(j+1)(k+1)-w(j-1)(k+1)) -(w(j+1)(k-1)-w(j-1)(k-1))]-L(ujk-Vxt)-Fxjk(1a) m2wjk/t2=K[wj(k+1)-2wjk+wj(k-1)]+eK[w(j+1)k-2wjk+w(j-1)k]+eK[(u(j+1)(k+1)-u(j+1)(k-1)) -(u(j-1)(k+1)-u(j-1)(k-1))]-L(wjk-Vyt)-Fyjk(1b) where xi=the position of the i-th slider, measured from its initial equilibrium position vi=the velocity of the i-th slider Vp=the plate moving speed m=the mass of a slider K=the strength of a coil spring L=the strength of a leaf spring Fo(qi,vi)=a velocity- and state-dependent friction force (with a fractal distribution of breaking strengths) qi=the state parameter of the i-th slider.

  19. Main Model Parameters 1. s=K/L: stiffness ratio (coupling factor) s>1: weakly coupling between the plate and the fault s<1: strongly coupling between the plate and the fault 2. gs=the decreasing rate of friction force with slip gv=the decreasing rate of friction force with velocity 3. g: the friction force drop factor (0<g<1) 4. Vp: the plate velocity (10-9 m/sec) 5. D: fractal dimension of the distribution of fault strengths 6. R: Roughness of fault strengths [=(Fsmax-Fsmin)/Fsmean] 7. m: the mass of a slider (inertial effect) (kg) 8. Density: volume density (kg/m3) and areal density (kg/m2)

  20. Boundary Conditions • Periodic BC: u1j=uNj (j=1, …, M); wi1=wiM (i=1, …, N) • Fixed BC: u1j=uNj=0 (i=1, …, N); wi1=wiM=0 (j=1, …, M) • (Stress-) Free BC: du1j/dx=duNj/dx=0 (j=1, …, M); dwi1/dy=dwiM/dy=0 (j=1, …, M); • Absorption BC: several ways • Mixed BC

  21. Incompleteness and Weakness of 1D and 2D Spring-slider Models 1. No seismic radiation term. (Exception: Xu and Knopoff (1994) used a radiation term like -aut) 2. How to exactly quantify the coupling effect? 3. How to exactly define the boundary condition? 4. Existence of finite-size effect (a finite number of sliders) 5. Numerical instability 6. The spring-slide model cannot be completely comparable with the classical crack.

  22. The Differential Equations Equivalent to the Difference Equations Dividing Eqs. (1a) and (1b) by dxdy leads to r2ujk/t2=K[u(j+1)k-2ujk+u(j-1)k]/dx2+eK[uj(k+1)-2ujk+uj(k-1)]/dy2+4eK[(w(j+1)(k+1)-w(j-1)(k+1))-(w(j+1)(k-1)-w(j-1)(k-1))]/4dxdy -L(ujk-Vxt)/dxdy-Fxjk/dxdy (2a) r2wjk/t2=K[wj(k+1)-2wjk+wj(k-1)]/dy2+eK[w(j+1)k-2wjk+w(j-1)k]/dx2+4eK[(u(j+1)(k+1)-u(j+1)(k-1))-(u(j-1)(k+1)-u(j-1)(k-1))]/4dxdy -L(wjk-Vyt)/dxdy-Fxjk/dxdy (2b) where r=m/dxdy is the areal density. Letting L=L/dxdy, fx=Fxjk/dxdy, and fy=Fyjk/dxdy and taking the limitation of dx and dygive 2u/t2=K2u/x2+eK2u/y2+4eK2w/xy-L(u-Vxt)-fx(3a) 2w/t2=K2w/y2+eK2w/x2+4eK2u/xy-L(w-Vyt)-fy(3b)

  23. General Forms of Solutions u(x,y,t)=u1e(ikr+Wt)+[Vxt-fx(0)]/L (4a) w(x,y,t)=w1e(ikr+Wt)+[Vyt-fy(0)]/L(4b) where k =<a, b>=vectorial wavenumber, w=angular frequency, and i=(-1)1/2. The scalar wavenumber is k=|k|. Inserting Eqs. (4a) and (4b) with r=<x, y> into Eqs. (3a) and (3b), respectively, leads to (W2+Ka2+eKb2+L-zW)u1+eKabw1=0 (5a) eKabu1+(W2+Ka2+eKb2+L-zW)w1=0 (5b) Eqs. (5a)–(5b) => Mx=0, where M is a 22 matrix of the coefficients, x is a 21 matrix of u1 and w1, and 0 is the 21 zero matrix.

  24. The condition for confirming the existence of solutions of Eq. (4) is |M|=0, i.e., (W2+Ka2+eKb2+L-zW)(W2+Ka2+eKb2+L-zW)-e2K2a2b2u1w1=0. This leads to 2W4-2hrW3+{[(1+e)Kk2+2L]+2z}W2-[h(1+e)Kk2+2Lz]W+eK2k2+(1+e)LKk2+L2=0. => W4+q3W3+q2W2+q1W+q0=0, where q3=-2h/, q2={[(1+e)Kk2+2L]+2z}/2, q2=-[h(1+e)Kk2+2Lz]/2, and q0=[eK4k4+(1+e)LKk2+L2]/2. On the basis of the Routh-Hurwitz theorem (cf. Franklin, 1968), four key parameters, i.e., n1, n2, n3, and n4, are taken to transform the expression R(W)=(W4+q2W2+d)/(q3W3+q1W) into the form R(W)=n1+1/[n2W+1/(n3W+1/n4W)].

  25. Im[W] Re[W] Mathematical manipulation leads to n1=1/q3, n2=q32/(q3q2-q1), n3=(q3q2-q1)2/q3(q3q2q1-q12-q32q0), and n4=(q3q2q1-q12-q32q0)/q0(q3q2-q1). The roots of R(W) all lie in the left half-side of the plane of Im[W] versus Re[W] if and only if all ni are positive. Obviously, n1>0. Since q3q2-q1=-z{[[(1+e)Kk2+2L]+2z2}/3<0, we have n2<0. This means that there is, at least, a root (say W*) of Eq. (5), whose real part appears in the right half-side of the plane of Im[W] vs. Re[W], that is, Re[W*]>0. Hence, u and w diverge with time in the form exp(Re[W*]t). Consequently, any small perturbation in the positions of the sliders, no matter how long or short its wavelength, will be amplified.

  26. Meanings of Model Parameters When L=0, and fx=fy=0, Eqs. (3a) and (3b), respectively, become 2u/t2=K2u/x2+eK2u/y2+4eK2w/xy (6a) 2w/t2=K2w/y2+eK2w/x2+4eK2u/xy (6b) The related wave equations in the 2-D space are rv2u/t2=(+2)2u/x2+2u/y2+(+)2w/xy (7a) rv2w/t2=(+2)2w/y2+2w/x2+(+)2u/xy(7b) where rV is the volume density with a dimension of mass per unit volume (e.g. kg/m3).

  27. From Eq. (7), the common P- and S-type wave velocities are [(+2)/rV]1/2 and (/rV)1/2, respectively. A comparison between Eq. (6) and Eq. (7) suggests that the P- and S- type wave velocities are (K/r)1/2 and (eK/r)1/2, respectively. Hence, related parameters are K=(+2)(r/rV), eK=(r/rV), e=(r/rV)/(+2), 4eK=(+)(r/rV), and e=(r/rV)(+)/4(+2). Obviously, L is not a function of elastic parameters of fault-zone materials.

  28. Areal Density • The mass of the cylinder is m=rVAh (rV=volume density). • The area density is defined to be r=m/A. This gives r=rVh. • Therefore, for the subsurface rocks the areal density increases with depth. m A h

  29. How to evaluate L?

  30. Conditions of Stable and Unstable Motions from One-body Single-degree-freedom Model Equation of Motion: m(d2d/dt2)=m(dv/dt)=te-tf. m: mass of the slider te(d)=k(do-d): elastic traction k: spring constant Vp: speed of loading point do: slip at the loading point (do=Vpt) tf : frictional stress v=dd/dt: Sliding velocity • A straight line with a slope of -L represents te=L(do-d) and crosses the t axis at t=to=LdL and the d axis at d=dL. • The te–d function with -L<-Lcr (or L>Lcr) cannot cross tf–d function, and thus te<tf. => a stable motion. • The te–d function with -L>-Lcr (or L<Lcr) crosses the tf–d function. From d=0 to the d at the intersection point, te>tf. => an unstable motion. • Hence, Lcr is the critical stiffness of the system. • For unstable motions, the inequality of gs >Lcr must hold. Hence, L<gs is the condition of generating an earthquake.

  31. logN=a-bM Homogeneous friction logN (N=Single frequency) Inhomogeneous friction Ml Ms Mc Md 0 Magnitude Physical Terms: x=s1/2= =(K/L)1/2 a=wpDo/2v1 wp=(L/m)1/2 Do=Fo/K u=v/wpDo v1=a characteristic velocity Carlson and Langer (Phys. Rev., 1989): Ms<M<Ml: microscopic events (sub-critical) M1<M<Mc: localized events (critical) Mc<M<Md: delocalized events (super-critical) Characteristic Magnitudes: Ms=ln[2pu(2s)-3/2] Mc=ln(2x/a) Md=ln(2L) => L=exp(Md)/2

  32. Seismic Coupling Coefficient, c • Definition: c=Mos,t/Mog,t where Mos,t is the seismic moment release rate of earthquakes and Mog,t is the moment rate estimated from geologically (or geodetically) measured fault slip rate (Peterson and Seno, JGR, 1984; Scholz and Campos, JGR, 1995) • For the Mariana arc: c=0.01 (weakly coupling=>smaller Mmax) • For the Chilean arc: c=1.57 (strongly coupling=>larger Mmax)

  33. Estimate of s from c • Mos=mfdfAf (mf=rigidity, df=slip, and Af=area in a fault zone) => Mos,t=mfAfdf,t • Mog=mgdgAg (mg=rigidity, dg=slip, and Ag=area around a fault zone) => Mog,t=mgAgdg,t • Since Af=Ag, c=Mos,t/Mog,t =mfdf,t /mgdg,t. • On the basis of the spring-slider model, df=-K(x-xo) and dg=-L(x-xo), and thus df,t～-Kvf (vf=slip velocity) and dg,t～-Lvg (vg=regional plate moving velocity). • This gives c=(mfKvf /mgLvg)=(mfvf /mgvg )s. • Hence, we have s=(mgvg/mfvf)c

  34. Angular Frequency and Phase Velocity The trial solutions are u~exp[i(kr-wt)] along the x-axis and w~exp[i(kr- wt)] along the y-axis. Since k=<a, b> and r=<x, y>, we have u~exp[i(ax+by-wt)] and w~exp[i(ax+by-wt)]. Inserting Eq. (6) the trial solutions results in (rw2-Ka2-eKb2)u-4eKabw=0 (8a) -4eKabu+(rw2-Kb2-eKa2)w=0 (8b) Eq. (8) => Mu=0, where M is a 22 matrix of coefficients and u is a 21 matrix of u and w.

  35. The condition for the existence of a non-trivial solution is |M|=0, i.e., 2w4-(1+e)K(a2+b2)rw2+eK2(a2+b2)2+K2[(1-e)2-16e2]a2b2=0 (9) Since k2=a2+b2 and (1-e)2-16e2=0, Eq. (9) becomes (w2)2-(1+e)Kk2(w2)+eK2k4=0 (10) The solution of Eq. (10) is w2=[(1+e)Kk2±(1-e)Kk2]/2 (11) For the “+” sign, let w=w1p and thus w1p2=Kk2. This leads to w1p=(K/)1/2k. The related wave velocity is C1p=w1p/k=(K/)1/2=[(+2)r/rV]1/2, which is constant and shows the P-type waves. For the “-” sign, let w=w1s and thus w1s2=eKk2. This leads to w1s=(eK/r)1/2k. It is obvious that w1p>w1s due to e<1. The related wave velocity is C1s=w1s/k=(eK/r)1/2= (r/rV)1/2=e1/2C1p, which is constant and exhibits the S-type waves.

  36. Types and Velocities of Propagating Waves LVxt and LVyt are only the loading stresses on a slider to make the total force reach its frictional strength. When they are, respectively, slightly higher than fox and foy, the slider moves and LVxt-fox and LVyt-foy are almost null and can be ignored during sliding. Hence, Eqs. (3a) and (3b) become, respectively, 2u/t2=K2u/x2+eK2u/y2+4eK2w/xy-Lu+gsu (12a) 2w/t2=K2w/y2+eK2w/x2+4eK2u/xy-Lw+gsw (12b) for slip-weakening friction, and 2u/t2=K2u/x2+eK2u/y2+4eK2w/xy-Lu+gvu/t(13a) 2w/t2=K2w/y2+eK2w/x2+4eK2u/xy-Lw+gvw/t (13b) for slip-weakening friction.

  37. Case 1: Coupling without friction For this case, L≠0 and fx=fy=0, Eqs. (8a) and (8b) or Eqs. (9a) and (9b), respectively, become (rw2-Ka2-eKb2-L)u-4eKabw=0 (14a) -4eKabu+(rw2-eKa2-Kb2-L)w=0 (14b) Eq. (14) =>Mu=0, where M is a 22 matrix of coefficients. The condition for the existence of a non-trivial solution is |M|=0, i.e., r2w4-[(1+e)K(a2+b2)+2L]rw2+{eK2(a2+b2)2+(1+e)KL(a2+b2) +K2[(1-e)2-16e2]a2b2+L2}=0 (15) Due to k2=a2+b2 and (1-e)2-16e2=0, Eq. (15) becomes r2w4-[(1+e)Kk2+2L]rw2+[eK2k4+(1+e)KLk2+L2]=0 (16)

  38. The solution of Eq. (16) is rw2={[(1+e)±(1-e)]Kk+2L}/2. Remarkably, coupling results in a constant increase in angular frequency and thus behaves like a low-cut filter. The related wave velocity, C, is C2=(w/k)2={[(1+e)±(1-e)](K/r)+2L/rk2}/2. For the “+” sign, let C=C2p and thus C2p=(C1p2+L/rk2)1/2 (17) The additional amount of wave velocity decreases with increasing k. When k>>1, C2p≈C1p. For finite k, C2p>C1p. Thus, this inequality and k-dependence of C2p show supersonic, dispersed P-type waves. When L=0, C2p=C1p. For the “-” sign, let C=C2s and thus C2s=(C1s2+L/rk2)1/2 (18) The additional amount of wave velocity decreases with increasing k. When k>>1, C2s≈C1s. For finite k, C2s>C1s. Thus, this inequality and k-dependence of C2s show supershear, dispersed S-type waves. When L=0, C2s=C1s.

  39. Case 2: Coupling and slip-weakening friction with a decreasing rate of gs exist L≠0 and fx=fy≠0 make Eqs. (12a) and (12b), respectively, become [rw2-Ka2-eKb2-(L-gs)]u-4eKabw=0 (19a) -4eKabu+[rw2-eKa2-Kb2-(L-gs)]w=0 (19b) Eq. (19) => Mu=0, where M is a 22 matrix of coefficients. The condition for the existence of a non-trivial solution is |M|=0, i.e., r2w4-[(1+e)K(a2+b2)+2(L-gs)]rw2+{eK2(a2+b2)2+(1+e)K(L-gs)(a2+b2) +K2[(1-e)2-16e2]a2b2+(L-gs)2}=0 (20)

  40. Due to k2=a2+b2 and (1-e)2-16e2=0, Eq. (15) becomes r2w4-[(1+e)Kk2+2(L-gs)]rw2+[eK2k4+(1+e)K(L-gs)k2+(L-gs)2]=0 (21) The solution of Eq. (21) is w2={[(1+e)±(1-e)]Kk2+2(L-gs)}/2r. Remarkably, coupling together with slip- weakening friction result in a constant change in angular frequency: an increase for L>gs, null for L=gs, and a decrease for L<gs. The related wave velocity is C=(w/k)2={[(1+e)±(1-e)](K/r)+2(L-gs)/rk2}/2. For the “+”sign, let C=C3p=[C1p2+(L-gs)/rk2]1/2. (22) The additional amount of wave velocity is dependent upon the difference between L and gs: positive for L>gs, null for L=gs, and, negative for L<gs. Its value decreases with increasing k. When k>>1, C3p≈C1p. It is noted that , L must be smaller than gs for producing faulting, and thus I have C3p<C1p. This inequality and k- dependence of C3p show subsonic, dispersed P-type waves. For the "-" sign, let C=C3s=[C1s2+(L-gs)/rk2]1/2 (23) The additional amount of wave velocity is dependent upon the difference between L and gs: positive for L>gs, null for L=gs, and, negative for L<gs. Its value decreases with increasing k. When k>>1, C3s≈C1s. It is noted that L must be smaller than gs for producing faulting, and thus C3s<C1s. This inequality and k-dependence of C3s show subshear, dispersed S-type waves.

  41. Case 3: Coupling and velocity-weakening friction with a decreasing rate of gs exist Inserting Eq. (13) the trial solutions leads to the following equations (rw2-Ka2-eKb2-L-igvw)u-4eKabw=0 (24a) -4eKabu+(rw2-eKa2-Kb2-L-iwgv)w=0(24b) Eq. (24) => Mu=0, where M is a 22 matrix of coefficients. The condition for the existence of a non-trivial solution is |M|=0, i.e., r2w4+2igvrw2-[r(Ka2+eKa2+Kb2+eKb2+L)+gv2] w2-igv[(eKa2+Kb2+L)+(Ka2+Kb2+L)]w+(Ka2+eKb2+L)(eKa2+Kb2+L2)-16e2K2a2b2=0 (25) Due to k2=a2+b2 and (1-e)2-16e2=0, Eq. (15) becomes r2w4-{r[(1+e)Kk2+2L]+gv2} w2+[eK2k4+(1+e)KLk2+L2] -i{2gvrw3+gv [(1+e)Kk2+2L]w}=0(26)

  42. Both the real and imaginary parts of Eq. (26) must be zero, i.e., r2w4-{r[(1+e)Kk2+2L]+gv2}w2+[eK2k4+(1+e)KLk2+L2]=0 (27a) 2gvrw3-gv [(1+e)Kk2+2L]w=0 (27b) For the real part, Eq. (27a) givesw2={[r(1+e)Kk2+2rL+gv2]{[r((1-e)Kk2+2L)+gv2]2- 4r[eKk4+ (1+e)KLk2+L2)]}1/2}/2r2. This leads to w2={[(C1p2+C1s2) k2+2L/r+(gv/r)2](C1p2-C1s2)2k2+2(C1p2+C1s2)(gv/r)2k2+[4L/r+(gv/r)2](gv/r)2}1/2/2. The wave velocity is C2=(w/k)2={[C1p2+C1s2+2L/rk2+(gv /rk)2]{(C1p2-C1s2) 2+ (C1p2+C1s2)(gv /rk) 2+[4L/rk2+(gv/rk)2](gv/rk)2}1/2. Since the terms inside the square root are all positive, C2 must be a real number. Obviously, the waves are composed of the P- and S-type waves. For the “+” sign, let C be C4p and thus C4p={[C1p2+C1s2+L/rk2+(gv/rk)2]+{(C1p2-C1s2)2+2(C1p2+C1s2) (gv/rk)2+[4L/rk2+(gv/rk)2](gv/rk)2}1/2}1/2/21/2 (28) C4p is a real number because all terms in Eq. (28) are positive. When k>>1, C4p≈C1p. For finite k, C4p>C1p. This inequality and k-dependence of C4p show supersonic, dispersed waves.

  43. For the “-”sign, let C be C4s and thus C4s={[C1p2+C1s2+2L/rk2+(gv/rk)2]-{(C1p2-C1s2)2+2(C1p2+C1s2) +(gv/rk)2 +[4L/rk2+(gv/rk)2 ](gv/rk)2}1/2}1/2/21/2 (29) Define u=C1p2+C1s2+2L/rk2+(gv/rk) 2 and q=(C1p2-C1s2) 2+2(C1p2+C1s2)(gv/rk) 2+ [4L/rk2+(gv/rk) 2](gv/rk) 2, thus giving u2-q=4[C1p2C1s2+(C1p2+C1s2)L/rk2+ (L/rk2) 2]=4(C1p2+L/rk2)(C1s2+L/rk2)=4C2p2C2s2>0. This gives u>q1/2, thus making C4s be a real number. When k>>1, C4s≈C1s. For finite k, C4s>C1s. This inequality and k-dependence of C4s show supersonic, dispersed waves. For the imaginary part, Eq. (27b), leads to another type of waves. Let w=w44 and thus w44={[(1+e)Kk2+2L]/2r}1/2=[(C1p2+C1s2) k2/2+L/r]1/2. The related wave velocity is C44=w44/k=[(C1p2+C1s2)/2+L/rk2]1/2 (30) This indicates that the waves are composed of the P- and S-type waves and independent of friction. However, the waves are different from those related to the real-part solutions. Eq. (30) suggests C44>C1s. The inequalities and k-dependence of C44 show non-causal, supersonic, dispersed waves.

  44. Table of Formulas

  45. The plots of C/Cmax versus T from 1 to 100 s: solid lines for C1p and C1s, dashed lines for C2p and C2s, upper dotted lines for C3p and C3s with gs=3×106 N·m-2/m, and lower dotted lines for C3p and C3s with gs=4×106 N·m-2/m under different values of L: (a) for L=1×104 N·m-2/m, (b) for L=2×104 N·m-2/m, and (c) for L=3×104 N·m-2/m when K=4.6×1014 N/m, r=2×107 kg/m2, and e=0.25. • Both C3p and C3s decrease with T and become zero when T is larger than a certain value which is dependent upon L and gs. • Inserting Eq. (22) for C3p and Eq. (23) for C3p, respectively, k=2p/TC1p and k=2p/TC1s leads to C3p=[1+(L-gs)T2/4rp2]1/2C1p and C3s=[1+(L-gs)T2/r4p2]1/2C1s. This gives C3p=0 and C3s=0 when T=2p[r/(gs-L)]1/2. Obviously, this characteristic period is the same for both P- and S-type waves. • When T>2p[r/(gs-L)]1/2, C3p and C3s become a complex number and thus the waves do not exist. Since gs must be larger than L for generating earthquakes, slip-weakening friction is not beneficial for producing longer-period waves.

  46. The plots of C/Cmax versus T from 1 to 100 s: solid lines for C1p and C1s, dashed lines for C2p and C2s, upper dotted lines for C4p and C4s with gv=1×106 N·m-2/m·s-1, and lower dotted lines for C4p and C4s with gv=2×106 N·m-2/m ·s-1 under different values of L: (a) for L=1×104 N·m-2/m, (b) for L=2×104 N·m-2/m, and (c) for L=3×104 N·m-2/m when K=4.6×1014 N/m, r=2×107 kg/m2, and e=0.25.

  47. Summary • There are only two types of waves for Cases 0, 1, and 2: One is the P-type wave and the other the S-type wave. • For Case 3, there are three types of waves, which are all composed of the P- and S-type waves. However, the first and second types of waves are, respectively, similar to the P- and S-type waves. The velocity of the third type of waves is always lower than the P-type wave velocity and higher than the S-type wave velocity. In other words, there are the subrsonic and supershear waves. • Coupling (for Cases 2 and 3) clearly increases the velocities of the two types of waves, thus leading to supersonic and supershear waves. • Slip-weakening friction for Case 2 decreases the velocities, thus only being able to result in subsonic and subshear waves. • When the period T>2p[r/(gs-L)]1/2, the waves do not exist for slip-decreasing friction, because gs must be larger than L for generating earthquakes. Hence, slip-weakening friction is not beneficial for producing longer-period waves. • Velocity-weakening friction makes the velocities of the first type of waves higher than the P-type wave velocity, while it makes the velocity of the second type of waves higher or lower than the S-type wave velocity just depending on the combination of L and gv.

  48. 謝謝 (Thanks)

More Related