Dynamical Spring-slider (Lattice) Dynamical Spring-slider (Lattice) Models for Earthquake FaultsModels for Earthquake Faults

Jeen-Hwa Wang, Jeen-Hwa Wang,

Institute of Earth Sciences, Academia SinicaInstitute of Earth Sciences, Academia Sinica

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

Viewpoints about a Fault Zone Viewpoints about a Fault Zone

• Geologists: A narrow zone with complex Geologists: A narrow zone with complex cataclastic deformationscataclastic deformations

• Rock Scientists: A narrow zone with gouge and Rock Scientists: A narrow zone with gouge and localized deformationslocalized deformations

• Seismologists: One or several double couples of Seismologists: One or several double couples of forces exerting on a well-defined ruptured planeforces exerting on a well-defined ruptured plane

• Physicists: A domain of first-order phase Physicists: A domain of first-order phase transitiontransition

• Mathematicians: ? (I do not know.)Mathematicians: ? (I do not know.)

Ingredients and Capability of Models Simulating Earthquake Faults

A Minimal Set of Ingredients:A Minimal Set of Ingredients:

• 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

Current Capability:Current Capability:

• 1. Model: modest (e.g. spring-slider model and crack model )

• 2. Constitution law of friction: incomplete

• 3. Initial condition: unknown

Models for Earthquake Faults

A Comprehensive Set of Models: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)

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-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(i,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(i,vi)=a velocity- and state-dependent friction force (nt) i=the state parameter of the i-th slider.

Velocity

F (friction force)

Fo: the static frictional force (Breaking strength)Fd: the dynamic frictional force

v

Fo

Fd

Classical Friction Law

(a-b)>0: strengthening or hardening

(a-b)<0: weakening or softening

5 cm/year

Evolution EffectDirect Effect

The factor a-b is a function of sliding velocity, temperature, loading rate etc.

Velocity- and State-dependent

Friction Law

Commonly-used Velocity- and State-dependent Friction Law

One-state-variable Velocity- and State-

dependent Friction Law:

=o+a(v/vo)+bln(vo/)

The laws describing the state variable, :

Slowness law: d/dt=-(v/)ln(v/)

Slip law: d/dt=-(v/)

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

Velocity

F (friction force)

rw rh

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 velocityrh: the increasing rate of friction force with velocity (healing of friction)

vc

Fo

gFo

Simplified Velocity-weakening Friction Law

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

Main Model ParametersMain 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)

Some Properties of the Spring-slider ModelSome 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*=2c/(b-a)max Lc: the characteristic size h>h*=> chaotic behavior h<h* => periodic behavior For the spring-slider models, h*=0 => chaotic behavior4. 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)

Three Rupture Modes in the 1-D Model(Wang, BSSA, 1996)

C2=Co2+[Kl/m-(rw/2m)2]/2

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)

S=50rw=1g=0.8

S=100rw=1g=0.8

S=50rw>>1g=0.6

S=100rw>>1g=0.6

D=1.5; R=0.5 Wang (1995)

Need a Two-dimensional Model

Ma et al. (2003)

• A 2-D dynamic model, with a more realistic constitution law of friction, is strongly needed for the studies of earthquakes and seismicity.

2-D N×M-body Dynamical Model(Wang, BSSA, 2000, 2012)

The equations of motion of the (i, j) slider are:

m2ujk/t2=K[u(j+1)k-2ujk+u(j-1)k]+eK[uj(k+1)-2ujk+u]+K[(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)

m2wjk/t2=K[wj(k+1)-2wjk+wj(k-1)]+eK[w(j+1)k -2wjk+w(j-1)k]+K[(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(i,vi)=a velocity- and state-dependent friction force (with a fractal distribution of breaking strengths)

i=the state parameter of the i-th slider.

Main Model ParametersMain 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. s=the decreasing rate of friction force with slip

v=the decreasing rate of friction force with velocity3. 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)

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

Incompleteness and Weakness of Incompleteness and Weakness of 1D and 2D Spring-slider Models 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.

The Differential Equations Equivalent to the Difference Equations

Dividing Eqs. (1a) and (1b) by xy leads to

2ujk/t2=K[u(j+1)k-2ujk+u(j-1)k]/x2+eK[uj(k+1)-2ujk+uj(k-1)]/y2

+4K[(w(j+1)(k+1)-w(j-1)(k+1))-(w(j+1)(k-1)-w(j-1)(k-1))]/4xy -L(ujk-Vxt)/xy-Fxjk/xy (2a)

2wjk/t2=K[wj(k+1)-2wjk+wj(k-1)]/y2+eK[w(j+1)k-2wjk+w(j-1)k]/x2

+4K[(u(j+1)(k+1)-u(j+1)(k-1))-(u(j-1)(k+1)-u(j-1)(k-1))]/4xy -L(wjk-Vyt)/dxdy-Fxjk/xy (2b)

where =m/xy is the areal density. Letting L=L/xy, fx=Fxjk/xy, and

fy=Fyjk/xy and taking the limitation of x and y give

2u/t2=K2u/x2+eK2u/y2+4K2w/xy-L(u-Vxt)-fx (3a)

2w/t2=K2w/y2+eK2w/x2+4K2u/xy-L(w-Vyt)-fy (3b)

General Forms of Solutions

u(x,y,t)=u1e(ir+t)+[Vxt-fx(0)]/L (4a)

w(x,y,t)=w1e(ir+t)+[Vyt-fy(0)]/L (4b)

where =<, >=vectorial wavenumber, =angular frequency, and i=(-1)1/2.

The scalar wavenumber is =||. Inserting Eqs. (4a) and (4b) with r=<x, y> into

Eqs. (3a) and (3b), respectively, leads to

(2+K2+eK2+L-)u1+Kw1=0 (5a)

Ku1+(2+K2+eK2+L-)w1=0 (5b)

Eqs. (5a)–(5b) => Mx=0, where M is a 22 matrix of the coefficients, x is a 21

matrix of u1 and w1, and 0 is the 21 zero matrix.

The condition for confirming the existence of solutions of Eq. (4) is |M|=0, i.e.,

(2+K2+eK2+L-)(2+K2+eK2+L-)-e2K222u1w1=0.

This leads to

24-23+{[(1+e)K2+2L]+2}2-[(1+e)K2+2L]+eK22+(1+e)LK2+L2=0.

=> 4+q33+q22+q1+q0=0,

where q3=-2/, q2={[(1+e)K2+2L]+2}/2, q2=-[(1+e)K2+2L]/2, and

q0=[eK44+(1+e)LK2+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()=(4+q22+d)/(q33+q1) into the form

R()=n1+1/[n2+1/(n3+1/n4)].

Mathematical manipulation leads to

n1=1/q3,

n2=q32/(q3q2-q1),

n3=(q3q2-q1)2/q3(q3q2q1-q12-q3

2q0), and

n4=(q3q2q1-q12-q3

2q0)/q0(q3q2-q1).

The roots of R() all lie in the left half-side of the plane of Im[] versus

Re[] if and only if all ni are positive.

Obviously, n1>0. Since q3q2-q1=-{[[(1+e)K2+2L]+22}/3<0, we

have n2<0. This means that there is, at least, a root (say *) of Eq. (5),

whose real part appears in the right half-side of the plane of Im[]

vs. Re[], that is, Re[*]>0. Hence, u and w diverge with time in the

form exp(Re[*]t).

Consequently, any small perturbation in the positions of the sliders,

no matter how long or short its wavelength, will be amplified.

Im[]

Re[]

Meanings of Model Parameters

When L=0, and fx=fy=0, Eqs. (3a) and (3b), respectively, become

2u/t2=K2u/x2+eK2u/y2+4K2w/xy (6a)

2w/t2=K2w/y2+eK2w/x2+4K2u/xy (6b)

The related wave equations in the 2-D space are

v2u/t2=(+2)2u/x2+2u/y2+(+)2w/xy (7a)

v2w/t2=(+2)2w/y2+2w/x2+(+)2u/xy (7b)

where V is the volume density with a dimension of mass per unit

volume (e.g. kg/m3).

From Eq. (7), the common P- and S-type wave velocities are

[(+2)/V]1/2 and (/V)1/2, respectively.

A comparison between Eq. (6) and Eq. (7) suggests that the P- and S-

type wave velocities are (K/)1/2 and (eK/)1/2, respectively. Hence,

related parameters are

K=(+2)(/V),

eK=(/V),

e=(/V)/(+2),

4eK=(+)(/V), and

=(/V)(+)/4(+2).

Obviously, L is not a function of elastic parameters of fault-zone

materials.

Areal Density

• The mass of the cylinder is m=VAh (V=volume density).

• The area density is defined to be =m/A. This gives =Vh.

• Therefore, for the subsurface rocks the areal density increases with depth.

m

A

h

How to evaluate L?

Conditions of Stable and Unstable Motions from One-body Single-degree-freedom Model

Equation of Motion: m(d2/dt2)=m(dv/dt)=e-f.m: mass of the slider

e()=(o-): elastic traction: spring constant

Vp: speed of loading point

o: slip at the loading point (o=Vpt)

f : frictional stressv=d/dt: Sliding velocity

• A straight line with a slope of -L represents e=L(o-) and crosses the axis at =o=LL and the axis at =L.

• The e– function with -L<-Lcr (or L>Lcr) cannot cross f–function, and thus e<f. => a stable motion.

• The e– function with -L>-Lcr (or L<Lcr) crosses the f–function. From =0 to the at the intersection point, e>f. => an unstable motion.

• Hence, Lcr is the critical stiffness of the system.

• For unstable motions, the inequality of s >Lcr must hold. Hence, L<s is the condition of generating an earthquake.

logN=a-bMlogN=a-bM

Magnitude

logN (N=Single frequency)

0 Md

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[2(2s)-3/2]

Mc=ln(2/)

Md=ln(2L) => L=exp(Md)/2

McMsMl

Homogeneous friction

Inhomogeneous friction

Physical Terms:

=s1/2= =(K/L)1/2

=pDo/2v1

p=(L/m)1/2

Do=Fo/K

=v/pDo

v1=a characteristic velocity

Seismic Coupling Coefficient,

• Definition: =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: =0.01 (weakly coupling=>smaller Mmax)

• For the Chilean arc: =1.57 (strongly coupling=>larger Mmax)

Estimate of s from • Mos=fdfAf (f=rigidity, df=slip, and Af=area in a fault zone)

=> Mos,t=fAfdf,t

• Mog=gdgAg (g=rigidity, dg=slip, and Ag=area around a fault zone) => Mog,t=gAgdg,t

• Since Af=Ag, =Mos,t/Mog,t =fdf,t /gdg,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 =(fKvf /gLvg)=(fvf /gvg )s.

• Hence, we have s=(gvg/fvf)

Angular Frequency and Phase Velocity

The trial solutions are u~exp[i(r-t)] along the x-axis and w~exp[i(r-

t)] along the y-axis. Since =<, > and r=<x, y>, we have

u~exp[i(x+y-t)] and w~exp[i(x+y-t)].

Inserting Eq. (6) the trial solutions results in

(2-K2-eK2)u-4Kw=0 (8a)

-4Ku+(2-K2-eK2)w=0 (8b)

Eq. (8) => Mu=0, where M is a 22 matrix of coefficients and u is a 21

matrix of u and w.

The condition for the existence of a non-trivial solution is |M|=0, i.e.,

24-(1+e)K(2+2)2+eK2(2+2)2+K2[(1-e)2-162]22=0 (9)

Since 2=2+2 and (1-e)2-162=0, Eq. (9) becomes

(2)2-(1+e)K2(2)+eK24=0 (10)

The solution of Eq. (10) is

2=[(1+e)K2± (1-e)K2]/2 (11)

For the “+” sign, let =1p and thus 1p2=K2. This leads to 1p=(K/)1/2. The

related wave velocity is C1p=1p/=(K/)1/2=[(+2)/V]1/2, which is constant and

shows the P-type waves.

For the “-” sign, let =1s and thus 1s2=eK2. This leads to 1s=(eK/)1/2. It is

obvious that 1p>1s due to e<1. The related wave velocity is C1s=1s/=(eK/)1/2=

(/V)1/2=e1/2C1p, which is constant and exhibits the S-type waves.

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=K2u/x2+eK2u/y2+4K2w/xy-Lu+su (12a)

2w/t2=K2w/y2+eK2w/x2+4K2u/xy-Lw+sw (12b)

for slip-weakening friction, and

2u/t2=K2u/x2+eK2u/y2+4K2w/xy-Lu+vu/t (13a)

2w/t2=K2w/y2+eK2w/x2+4K2u/xy-Lw+vw/t (13b)

for slip-weakening friction.

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

(2-K2-eK2-L)u-4Kw=0 (14a)

-4Ku+(2-eK2-K2-L)w=0 (14b)

Eq. (14) =>Mu=0, where M is a 22 matrix of coefficients. The condition

for the existence of a non-trivial solution is |M|=0, i.e.,

24-[(1+e)K(2+2)+2L]2+{eK2(2+2)2+(1+e)KL(2+2) +K2[(1-e)2-162]22+L2}=0 (15)

Due to 2=2+2 and (1-e)2-162=0, Eq. (15) becomes

24-[(1+e)K2+2L]2+[eK24+(1+e)KL2+L2]=0 (16)

The solution of Eq. (16) is 2={[(1+e)± (1-e)]K+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=(/)2={[(1+e)± (1-e)](K/)+2L/2}/2.

For the “+” sign, let C=C2p and thus

C2p=(C1p2+L/2)1/2 (17)

The additional amount of wave velocity decreases with increasing . When >>1,

C2p≈C1p. For finite , C2p>C1p. Thus, this inequality and -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/2)1/2 (18)

The additional amount of wave velocity decreases with increasing . When >>1,

C2s≈C1s. For finite , C2s>C1s. Thus, this inequality and -dependence of C2s show

supershear, dispersed S-type waves. When L=0, C2s=C1s.

Case 2: Coupling and slip-weakening friction with a decreasing rate of s exist

L≠ 0 and fx=fy≠0 make Eqs. (12a) and (12b), respectively, become

[2-K2-eK2-(L-s)]u-4Kw=0 (19a)

-4Ku+[2-eK2-K2-(L-s)]w=0 (19b)

Eq. (19) => Mu=0, where M is a 22 matrix of coefficients. The

condition for the existence of a non-trivial solution is |M|=0, i.e.,

24-[(1+e)K(2+2)+2(L-s)]2+{eK2(2+2)2

+(1+e)K(L-s)(2+2) +K2[(1-e)2-162]22+(L-s)2}=0 (20)

Due to 2=2+2 and (1-e)2-162=0, Eq. (15) becomes

24-[(1+e)K2+2(L-s)]2+[eK24+(1+e)K(L-s)2+(L-s)2]=0 (21)

The solution of Eq. (21) is 2={[(1+e)± (1-e)]K2+2(L-s)}/2. Remarkably, coupling together with slip-

weakening friction result in a constant change in angular frequency: an increase for L>s, null for L=s, and a

decrease for L<s. The related wave velocity is

C=(/)2={[(1+e)± (1-e)](K/)+2(L-s)/2}/2.

For the “+” sign, let C=C3p=[C1p2+(L-s)/2]1/2. (22)

The additional amount of wave velocity is dependent upon the difference between L and s: positive for L>s,

null for L=s, and, negative for L<s. Its value decreases with increasing . When >>1, C3p≈C1p. It is noted

that , L must be smaller than s for producing faulting, and thus I have C3p<C1p. This inequality and -

dependence of C3p show subsonic, dispersed P-type waves.

For the "-" sign, let C=C3s=[C1s2+(L-s)/2]1/2 (23)

The additional amount of wave velocity is dependent upon the difference between L and s: positive for L>s,

null for L=s, and, negative for L<s. Its value decreases with increasing . When >>1, C3s≈C1s. It is noted

that L must be smaller than s for producing faulting, and thus C3s<C1s. This inequality and -dependence of

C3s show subshear, dispersed S-type waves.

 

Case 3: Coupling and velocity-weakening friction with a decreasing rate of s exist

Inserting Eq. (13) the trial solutions leads to the following equations

(2-K2-eK2-L-iv)u-4Kw=0 (24a)

-4Ku+(2-eK2-K2-L-iv)w=0 (24b)

Eq. (24) => Mu=0, where M is a 22 matrix of coefficients. The

condition for the existence of a non-trivial solution is |M|=0, i.e.,

24+2iv2-[(K2+eK2+K2+eK2+L)+v2]2

-iv[(eK2+K2+L)+(K2+K2+L)]+(K2+eK2+L)(eK2+K2+L2)-162K222=0 (25)

Due to 2=2+2 and (1-e)2-162=0, Eq. (15) becomes

24-{[(1+e)K2+2L]+v2}2+[eK24+(1+e)KL2+L2]

-i{2v3+v

[(1+e)K2+2L]}=0(26)

Both the real and imaginary parts of Eq. (26) must be zero, i.e.,

24-{[(1+e)K2+2L]+v2}2+[eK24+(1+e)KL2+L2]=0 (27a)

2v3-v [(1+e)K2+2L]=0 (27b)

For the real part, Eq. (27a) gives 2={[(1+e)K2+2L+v2]{[((1-e)K2+2L)+v

2]2-

4[eK4+ (1+e)KL2+L2)]}1/2}/22. This leads to 2={[(C1p2+C1s

2)

2+2L/+(v/)2](C1p2-C1s

2)22+2(C1p2+C1s

2)(v/)22+[4L/+(v/)2](v/)2}1/2/2. The

wave velocity is C2=(/)2={[C1p2+C1s

2+2L/2+(v /)2]{(C1p2-C1s

2) 2+

(C1p2+C1s

2)(v /) 2+[4L/2+(v/)2](v/)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+C1s

2+L/2+(v/)2]+{(C1p2-C1s

2)2+2(C1p2+C1s

2) (v/)2+[4L/2+(v/)2](v/)2}1/2}1/2/21/2 (28)

C4p is a real number because all terms in Eq. (28) are positive. When >>1, C4p≈C1p.

For finite , C4p>C1p. This inequality and -dependence of C4p show supersonic,

dispersed waves.

For the “-” sign, let C be C4s and thus

C4s={[C1p2+C1s

2+2L/2+(v/)2]-{(C1p2-C1s

2)2+2(C1p2+C1s

2) +(v/)2 +[4L/2+(v/)2 ](v/)2}1/2}1/2/21/2 (29)

Define =C1p2+C1s

2+2L/2+(v/) 2 and =(C1p2-C1s

2) 2+2(C1p2+C1s2)(v/) 2+

[4L/2+(v/) 2](v/) 2, thus giving 2-=4[C1p2C1s

2+(C1p2+C1s

2)L/2+

(L/2) 2]=4(C1p2+L/2)(C1s

2+L/2)=4C2p2C2s

2>0. This gives >1/2, thus

making C4s be a real number. When >>1, C4s≈C1s. For finite , C4s>C1s. This

inequality and -dependence of C4s show supersonic, dispersed waves.

For the imaginary part, Eq. (27b), leads to another type of waves. Let =44 and thus

44={[(1+e)K2+2L]/2}1/2=[(C1p2+C1s

2)2/2+L/]1/2. The related wave velocity is

C44=44/=[(C1p2+C1s

2)/2+L/2]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.

P-type Waves S-type Waves Other

Case 0L=0F=0

C1p=(K/)1/2=[(+2)/V]1/2 C1s=(eK/)1/2=(/V)1/2=e1/2C1p

Case 1L≠0F=0

C2p=(C1p2+L/2)1/2 C2s=(C1s

2+L/2)1/2

Case 2L≠0F≠0 (slip-dependent)

C3p=[C1p2+(L-s)/2]1/2 C3s=[C1s

2+(L-s)/2]1/2

Case 3L≠0F≠0 (velocity-dependent)

C4p={[C1p2+C1s

2+L/2+(v/)2]+{(C1p2-

C1s2)2+2(C1p

2+C1s2)(v/)2+

[4L/2 (v/)2](v/)2}1/2}1/2/21/2

C4s={[C1p2+C1s

2+2L/2+(v/)2]-{(C1p

2-C1s2)2+2(C1p

2+C1s2)+(v/)2

+[4L/2+ (v/)2](v/)2}1/2}1/2/21/2

C44=[(C1p2+C1s

2)/2+L/2]1/2

Table of FormulasTable of Formulas

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 s=3×106 N·m-2/m, and lower dotted lines for C3p and C3s with s=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, =2×107 kg/m2, and =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 s.

• Inserting Eq. (22) for C3p and Eq. (23) for C3p, respectively, =2/TC1p and =2/TC1s leads to C3p=[1+(L-s)T2/42]1/2C1p and C3s=[1+(L-s)T2/42]1/2C1s. This gives C3p=0 and C3s=0 when T=2[/(s-L)]1/2. Obviously, this characteristic period is the same for both P- and S-type waves.

• When T>2[/(s-L)]1/2, C3p and C3s become a complex number and thus the waves do not exist. Since s must be larger than L for generating earthquakes, slip-weakening friction is not beneficial for producing longer-period waves.

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 v=1×106 N·m-2/m·s-1, and lower dotted lines for C4p and C4s with v=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, =2×107 kg/m2, and =0.25.

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>2[/(s-L)]1/2, the waves do not exist for slip-decreasing friction, because s 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 v.

謝謝 (Thanks)