A plaquet representation of ruptures and models for earthquakes

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A plaquet representation of ruptures and models for earthquakes

Huang-Jian Xu, Birger Bergersen, Kan Chen

To cite this version:

Huang-Jian Xu, Birger Bergersen, Kan Chen. A plaquet representation of ruptures and models for earthquakes. Journal de Physique I, EDP Sciences, 1993, 3 (10), pp.2029-2040. �10.1051/jp1:1993230�.

�jpa-00246850�

Classification Physics Abstracts

46.30N 05.50 91.308

A plaquet representation of ruptures and models for earthquakes

Huang-Jian

Xu (~>*),

Birger Bergersen (I)

and Kan Chen

(2)

(~) Department of Physics, University of British Columbia Vancouver, B-C- V6TlZl, Canada (~) Department of Physics, National University of Singapore, Singapore 0511

(Received

6 May1993, accepted 8 June

1993)

Abstract. We present a model for ruptures in _an elastic medium. The basic unit is _a

plaquet _on which local strain and stress tensors _are assigned. We show that _our discretization naturally leads to _a double-couple representation of local ruptures. The stress redistribution due to ruptures is conveniently calculated using lattice Green's functions. This description _can be

immediately applied to the study of earthquakes, which _{are a sequence} of ruptures generated by

continuous stress increase in _a medium. We show that _our model is suitable for incorporating

some basic properties of rocks

(e.g.

static fatigue and healing of the fractured

surfaces),

_as well

as for studying the spatial patterns of earthquake activities.

1. Introduction.

In _an earlier _paper

ill

_we introduced _a discretization of continuum elastic theory with _a double-

couple representation

of local ruptures. We

applied

this discretization

technique

to the

study

of

earthquakes

and demonstrated how the tensor character of the stresses can be taken into account in _a

dynamical

model. We also showed how to

incorporate

some basic

properties

^of

rocks

(e.g.

static

fatigue

and

healing

of the fractured

surface)

in the local

dynamics

of the model. The _purpose of the present _paper is

(I)

to describe the details of the discretization and lattice Green's function

approach

for

calculating

the stress redistribution due to ruptures ^and

(it)

to present a

study

of the

earthquake

model and discuss various

phenomenological

laws of

earthquakes generated

in _our models.

One of the

major challenges

^{of material} science research is to understand the breakdown process of _a

heterogeneous

solid. This is not

only technologically

important, but also crucial in

understanding

natural

phenomena

such _as

earthquakes.

There _{are a} few standard

approaches

in the

study

of this fracture _process. Molecular

dynamics

_can simulate the formation and

propagation

of _a crack _{on a} microscopic level. The

method,

however,

only

has limited

appli-

cations, _as it is

computationally

too

demanding

to

study

the medium with _more than _a few (*) Present Address Institute of Geophysics and Planetary Physics, University of California, Los Angeles CA 90024, U-S-A--

cracks. A formal

approach widely

used in

engineering

is the finite element

method,

due to its

simplicity

^and

flexibility

in

dealing

with very different

problems. However,

the

description

of

a real elastic solid will be accurate

only

if the

displacement

field varies

slowly

_over the size of the elements used. If there is crack in _a

medium,

the

displacement

field around the crack will be accurate

only

if the elements _are much smaller than the size of the crack itself.

Also,

in

modeling

_a

heterogeneous solid,

the size of the

heterogeneities

should be

larger

than the size of the elements. This is _a serious drawback if _one deals with many microcracks in _a disordered

medium.

Recently,

_a few

simple

lattice

models,

_e.g. the central force model and the beam

model,

have been

proposed by physicists.

For _a review of these

models,

_see reference [2]. ^In

these

approaches,

the medium consists of _a network of bonds

(or beams)

and _a local rupture is

simply

the

breaking

of _a bond. The

approach

_{we use} is different. The basic unit of _our dis-

cretization is _a

plaquet;

this has the

advantage

that the stress and strain tensors can be defined

directly. Also,

local

physics

such _as static

fatigue, healing,

etc. _can be

incorporated

rather

naturally

in _our model. This is

particularly

useful for

modeling

the

dynamics

of

earthquakes,

which _we will discuss _{as an}

application

of _our discretization

approach.

Earthquakes

_can be viewed _{as sequences} of ruptures ^which release the elastic stress built _up

gradually by

the movement of tectonic

plates.

The

global dynamics

of

earthquakes

is

quite

rich and

complex.

There have been many attempts at

earthquake modeling,

yet there _are still many

unsolved

problems.

Part of the _reason is the

large

amount of

phenomenology

needed _to

model,

a more important reason is that _one does not know how to model the

physics

well. One of the

outstanding problems

is to

incorporate

the tensor character of the stresses

satisfactorily

into _a

dynamical

model of

earthquakes

_[3]. This is the

problem

_we attempt to solve in _our model of

earthquakes.

We also show that _a few

empirical

laws

regarding healing

^of fractured surfaces [4]

(for ensuring

_a

steady state)

and static

fatigue

of rocks [5, 6]

(responsible

for

generating aftershocks)

_can be

incorporated naturally

in _our model. In addition _to

reproducing

_a number of

phenomenological

laws such _as the

Gutenberg-llichter

law

iii (as

in _a number of

previous

earthquake

models

[8-14]),

_our model _can also

dynamically _generate

_a

relatively

stable fault structure, ^made up of

regions

that have been weakened

by earthquakes.

2. Lattice

representation

of ruptures in _an elastic medium.

In this section _we will concentrate on a detailed

description

^of _our discretization

procedure

and the

representation

of local ruptures. The first step is to construct _a discrete derivative.

For

simplicity

let _us restrict _our attention to ruptures ^due to shear stress in the _x-y

plane.

Generalization to three dimensions is

straightforward.

In _our work _{we use} the

following

forms of the discretized derivative

(say

for _a function

g(r)):

D«g(r)

+

(lg(r

⁺

^b)

⁺

^g(r

⁺

^{d) g(r b) g(r d)i, (1)}

and

Dyg(r)

₊

ig(r

+

b) g(r

+

d) g(r b)

₊

g(r d)j, (2)

where

dz,y

are unit vectors

(the

lattice

spacing

is taken _to be

unity),

d

=

(dz dy)/2

and b =

(lx

+

dy)/2.

Note that if _g sits _on the nodes of _{a square}

lattice,

^then

Dzg

sits _on the centers of the

plaquets

formed

by

the nodes of this _square

lattice,

and vice _versa. A consequence of this definition is that the basic unit of _our discretization is _a

plaquet

rather than _a bond.

Once

Dz,yg

is

defined,

the discretization of elastic

theory

is

straightfomard.

First _we define

a

displacement

vector _{u on} each node

(corner

of _a

plaquet).

The distortion of the

plaquet

is

then characterized

by

the strain tensor

(defined

at the _center of each

plaquet) U,j(r)

=

((D,Uj(r)

⁺

^{Djv;(r)). (3)}

When all deformation _are elastic

(no

ruptures

exist)

the _{stress tensor} is related _to the strain tensor

through

the

generalized

Hooke's law:

air(r)

"

XVII(r)dir

+

2P(UIJ(r) )diJUii(r)), ⁽⁴⁾

where

K,

_{p are} bulk and shear

module, respectively,

and summation _over

repeated

indices is

implied.

The discretized elastic

equation

is

given by

the force balance

condition, which,

in discrete

form,

is

F; =

Dja;j

₌ 0.

(5)

We further restrict _our consideration to the shear mode of rupture

(most earthquakes

_are

shear mode

fractures).

In this _case

only

two components of the stress _are needed in order to determine where _a rupture will _{occur: ai}

" azy the shear stress at the

dz,y direction,

and

a2 "

(ayy azz)/2

the shear stress at the directions of b _or d. In this paper we

only

consider ruptures in the

f, j

and b _or d directions. The ruptures ⁱⁿ an

arbitrary

direction _can be

represented

_{as a} linear combination of these two _cases.

(The

shear stress in the direction

which makes _an

angle

with x-direction is

simply

_{ai cos} 2H _{+ a2} sin 2H).

Let _us consider the effect of _a local rupture in _an infinite

medium,

_{say a} shear fracture in the lx direction at

plaquet

centered at ro. After the break the

original

shear _{stress on} the

fractured surface is reduced to _a percentage x of the

original

stress. The stress reduction _causes

a force imbalance and _a

subsequent

_stress redistribution _occurs. We denote the stress after the rupture as

a[~~

^{and the} stress before the rupture as

a$(~.

^{We have}

a[5~

₌

a[(~

+

a(~,

^where a(~ ^{is the additional stress caused}

by

the rupture. ^At

equilibrium a[~~

must also

satisfy

the

force-balance

equation,

and _we must have

Dj@(~ " 0.

(6)

Let

u((r)

denote the additional

displacement

induced

by

the rupture. a(~ can be related _to

u(

via the

generalized

Hooke's law

throughout

^the system except ^for

a[~(ro)

at the

ruptured

site. To separate the violation of Hooke's law at the

ruptured

site, _we write

~ly(~)

"

~l[(~)

+

~~~dr,ro, (7)

~lz(~)

"

~ll(~), (8)

~~Y~~~ ~~~~~~' ~~~

where

a])

^is the part of ^{stress that is} related to

u(

^via the

generalized

Hooke's law.

Substituting

these

expressions

in

(6)

_we obtain the

equations

for a~~.

Dja[)(r)

+

a"~Dy(dr,ro)

"

0, (10)

Dja(((r)

₊

a"~Dz(dr,ro

" 0.

(ll)

Given a"~ we can obtain _a solution for a~~, ^{which then} can be used to obtain a"~ self-

consistently by applying

the

boundary

condition for the

ruptured

site

a](~(ro)

"

xa](~(ro) ⁽¹²⁾

f

f la)

16)

I

Fig-I-

Schematic illustration of _a region subjected to external shear stress. The system is divided into many plaquets, and is driven by slowly increasing shear stress. When _a local stress is larger than the corresponding threshold stress, the plaquet fractures, causing _a long-range redistribution of elastic forces. The force distribution of local double-couple _{sources are} also illustrated:

(a)

fracture in d~ _or iv direction;

(b)

fracture in d~ + iv direction.

I-e- the stress is reduced to _a fraction _x of the

original

stress.

The equation for a~~ _can be cast

(substituting

a"~ _e

-vsf)

_as

Dja[(

_{F,~~ =} 0,

(13)

where

~f~

"

j (dr,ro+b dr,ro+d dr,ro-b

+

dr,ro-dj, ⁽¹⁴⁾

and

~l~

_"

j(dr,ro+b ^{dr,ro-d dr,ro-b}

⁺

^{dr,ro+dj. (15)}

Therefore,

a~~

can be viewed _as

generated by

the external _source

Fdb,

which is

only

nonzero at

the _corners of the fractured

surface,

and is shown in

figure

I. This force distribution satisfies

the condition that its net force and net torque are zero, and thus is _a double

couple (a

term used in

seismology

to describe the

earthquake source).

Our discretization is in

complete

agreement

with the well-established fact that _a shear rupture can be modeled

by

_a

double-couple

force redistribution [15].

Next,

_we want to determine

a[j, given

this

double-couple

force distribution. The solution of this

problem

_can be

easily

obtained in Fourier _space. Since the side of the unit cell is

unity

_we define the Fourier transform _as

g ~

l ^{2« 2«}

(27r)2

~

dk«

dkyg(k)elk.r (16)

The Fourier transform of the discrete derivative is then

given by

(D«g)(k)

= 2i

sin(k«/2) cos(k~/2)g(k)

_e

D~(k)g(k), (ii) (D~g)(k)

=

21sin(k~/2) cos(k«/2)g(k)

e

D~(k)g(k). (18)

Combining equations (3)

and

(4)

and

replacing

_a with a~~ and _u with u' _we obtain

Dj a[(

₌

[(K

+

4p/3)D(

₊

_pD(]u[

₊

(K

+

p/3)DzDyu[, (19)

and

Dja((

=

[(K

+

4p/3)D(

+

pD(]u[

+

(K

+

p/3)DzDyi1[. (20)

Fourier transformation of

equation (13) yields

1(K

+

4p/3)Dl

₊

pD]lUl(k)

₊

(K

+

p/3)D«DyU[(k) V5fDye~'~'~°

= 0,

(21)

and

[(K

+

4p/3)D(

+

pD(]u[(k)

+

(K

+

p/3)D~Dyu[(k) v5fDze~'~'~°

= 0.

(22)

These _are

coupled

linear

equations

and _can be solved

straightforwardly.

The solution is

vi (k)

=

i(K

+

4P/3)Di (K 2P/3)DiDyle~'~'~°, (23)

where

A(k) =

~el (~)

= K

2/l/3)(DzU[ (k)

+

DyU((k))

+ 21LDz~l~

(k)

_~~~~

~~

4DzD(

_{_,k_ro}

"

~ (26)

(Dj

+

Dj)2

^{~ '}

~jj(~)

=

(K 2p/3)(Dzu[(k)

₊

Dyu[(k))

+

21LDyU~(k)

(~~~

"

~(i~~)2~ ^'~~°'

~~~~

and

a(((k)

=

p(D~u[(k)

+

Dyu[ (k)) (29)

"

~/( ~~ _~)2

~

'~ _~°

+

v~f~

^'~

~°'

_(~°~

where

/

=

fyi(K

₊

~1/3)/(K

+

4p/3).

We

only

monitor the shear stresses ai and _a2. The additional shear _{stresses are}

given by

al(k)

=

aly(k)

=

al[(k)

₊

al(e~'~'~°, (31)

ai(k)

=

)(aiy(k) ^ai«(k))

=

)(ail(k) ^{all(k)). (32)}

Using

the fact that

a](

=

-Vif

_we obtain

4D(D(

_{_,k.ro}

al(k) (33)

"

-f

(Dj

₊

Dj)2~

~~~

,

-2DzDy(D( D()

~_;k.ro ₍₃₄₎

a2(~)

"

f

J~2 ₊

JJ2)2

In real _{space we} have

a[(r)

=

/Gi(r _{ro), (35)}

and

U~(~) "

~/G2(r _r0), (36)

where Gi and G2 _are ^{lattice Green's functions:}

~p ~~~

f"

^dk«

f"

^{dkY SIU~k«SIU~ kY}

^~ik.r

_~~~~

~

_~

27r

_~

27r

(1

_cos kz _cos ky )2 ^'

~

j"

^dkz

j" ^dky

^{sin kz sin}

^ky(cos

^{kz cos}

^ky)

^;k_r _~~~

~~~~

_~

27r

_~

27r

(1

cos kz _cos

ky

)2 ^~

If _{we assume} the

boundary

condition at the

ruptured

site, that after the rupture ^{the shear}

stress ai on the fractured segment ^is reduced to _a percentage x of the

original value,

then

/

can be determined

by

_ao +

a[(ro)

_{" Tao,} where _ao is the

original

shear stress

just

^{before the} rupture. ^{This leads} to

/Gi ₍₀₎

₊

_{(1 x)ao}

" 0, and

a[, a[

_can be written _as,

U~,2(~) "

~(l ~)U0Gl,2(r r0)/Gl (°). (~~)

We have checked that the Green functions reduce _{to correct} continuum limit for the double-

couple

stress redistribution. At

long

distances the Green functions

decay

_as

I/r~

in _our two dimensional

model,

where d is the

spatial

dimension. At short distances the greatest stress increase is _near the tip ^of a crack.

For _a fracture in the

I

or

I

_{direction the}

double-couple

force distribution is shown in

plaquet (b)

of

figure

I, and is

given

below,

F~(r)

=

fvs[-dd~

~ ~b

bd~

~ ~d ⁺

dd~

~ _b +

bd~

~

d]. (40)

We _can obtain

similarly

the additional stress

a[

~ due to _a rupture in the direction of

I

or

I, ai(r)

=

-(i x)aoG2(r ro)/(i Gi(0)), (41)

ai(r)

=

-(i x)ao(dr,o Gi(r ro))/(i Gi(0)). (42)

Again,

_ao is the

original

shear stress a2 at the

ruptured

site

just

before the rupture.

We _can also calculate the stress redistribution when _more than _one site ruptures. For sim-

plicity,

let _us consider the _case that all ruptures occur in ix direction. Let _ri, I

= I,

,

Nb

(Nb

is the total number of ruptures ⁱⁿ the

system)

be the locations of ruptures.

Again

we

can represent the ruptures

by double-couple

force distribution. We _assume the

corresponding strength

of the double

couples

be

fi,

I

= I,

,

Nb. Because of the crack-crack interaction _we

can not determine

fi

the _{same way as} in the _case of _a

single

rupture.

Instead,

the

boundary

condition

a[~~(ri)

"

xa/'~(ri) corresponds

to a set of linear equations which have to be solved

to obtain

fir

(1

x)a('~(ri) ~ /mGi _(ri rm)

= 0,

(43)

m

for = I,

,

Nb.

,After fi

is obtained _{we can} determine the additional stress a(~ ^caused

by

the ruptures ^{in the entire medium.}

,'

o

-ii

i oo

0 50 loo 150 200

Fig.

2. _{The shear stress a~y} redistribution calculated from the Green's function method in the text.

The length of the fault at the center is 100 lattice units. The shear stress is reduced by loo units

uniformly along the fault. This distribution is in good agreement with the analytical results

(see,

for

example, Fig. 8 in Ref. [16]).

As _a test

study

_we consider the stress redistribution caused

by

_{a segment} of consecutive _rup- tures in d~ direction

(with

_x

= 0

corresponding

to complete stress

release).

The

corresponding

continuum _{case a} segment ^{in the interior of the medium is}

ruptured

and the shear stress _on it reduced to _{zero can} be solved

analytically

_[16]. In

figure

2 _we

plot

the additional _stress

a[ generated by

the ruptures ^{from the}

procedure

described above. The agreement is excellent

(see

_e.g.

Fig.

8 of Ref.

[16]).

^This

again gives

_us ^confidence that _our discretization is _a very

good representation

of the elastic

theory.

3.

Applications

to

earthquakes.

We _now discuss the

application

of _our discretization scheme to

earthquake dynamics.

The

model _we

study

in this paper is the _{same as} the model _we

presented

in _our

previous

_paper except that in the present ^version we also

incorporate

_an

empirical

law

describing

the

healing

of _a fractured surface. In this section _we will present our model in _more detail.

In order to describe _a

dynamical

model of

earthquakes,

several

ingredients

_{are necessary.}

First,

_we need _a fracture criterion. This _can be achieved

by assigning

a distribution of

breaking

thresholds

(when

the local shear stress is

larger

than the

threshold,

_a rupture will

occur).

After each rupture,

however,

the fractured

plaquet

is weakened: the _stress threshold is reduced to _a lower value _Hi-

(Initially

_we

assign

the thresholds to be random numbers between _Hi and Hu).

In

reality

the threshold for shear failure of

previously

fractured surface is better described

by

H, =

of

+ van, where _an is the normal stress,

of

represents cohesion in the

material,

and _{v can} be

interpreted

_as the internal friction coefficient. As _a first step _we

only

consider the

special

case v = 0 in this paper. The

weakening

^{of fractured surfaces} is

responsible

for

generating

_a

fault _zone. In order that _a

steady

state can be

established,

the fractured surface is allowed to anneal

slowly.

The

healing

_process is not well understood. In this _{paper, we assume [4]} that the

strength

^{of the fractured surface} will grow as

H, = Hi + A

log(t t,), (44)

where t, denotes the time of the last fracture and _Hi is the lower cutoff of the threshold

(to

indicate the fact that after the rupture, the fractured surfaces _{are most}

likely

^still in contact, parts are not

completely ruptured,

and the fractured surface will be able to support _a small

applied stress).

The

healing

takes effect after _a

given

event. The

proposed healing

law is

taken _as

purely empirical

and other forms could

equally

well be proposed _11, 9]. ^In practice

healing

will take

place

_{over a}

hierarchy

of different

timescales,

equation

(44)

then takes into

account that there is _no

single

characteristic time for

healing. Second,

_we need to choose

boundary

conditions _{as we}

only

_can simulate _a finite

region.

In

earthquakes

it makes _sense to use open

(absorbing) boundary

conditions: _we consider _an infinite

medium,

but

neglect

the effect of ruptures outside the

region

_on which _we concentrate. This also has _an

advantage computationally,

because _{we can use} the Green's functions for the infinite medium.

Third,

in order to obtain

earthquake

_sequences in the model

containing aftershocks,

_some time

delay

mechanism for ruptures must also be

included,

this will be discussed later.

The system ^{is driven}

by uniformly increasing

the shear stress at a constant rate p, with _zero initial stress.

Eventually

the stress on one of the

plaquets

will exceed its threshold and it will rupture. The stress redistribution from this rupture is

computed

and _we check if this _causes

the stress on other

plaquets

to exceed their thresholds. This

produces

_a list of

plaquets

which

will rupture in the next step. ^In the simulations

presented

here the stress redistribution from these

plaquets

_are

simply

calculated

individually

and

summed,

rather than

using (43).

This process is the _{same as was} used in [14]. ^{It could be mentioned}

that, although

the time scale for

breaking

and stress redistribution _are

fast,

these _{processes are not}

truly

instantaneous. One could

easily implement

different selection schemes for the order in which the

plaquets

break.

The scheme used _assumes that the time for restoration of mechanical

equilibrium

is much shorter than other time scales in the

problem

equations are satisfied _we do not believe that these details _are

important

for the overall

picture.

In the calculations

presented

here _we restrict ourselves to the _case where the stress is in- creased for the component _a2. As discussed elsewhere [17], a

large

fraction of the events take

place

_near the

edges,

if _we instead

uniformly

increase the component _a~y in _a square

region

with sides

parallel

to _x and y. We

only

consider the shear mode of rupture and

begin by

considering

the model without static

fatigue.

We _measure the size of _an

earthquake by

the total number _s of sites that have

ruptured

following

the initial

instability.

If _a

given

site ruptures more than _once

during

_{an event every}

occurrence is counted. We also define the "seismic moment" which is

proportional

to the

summed stress

drop

_over the

ruptured

sites. We have calculated the distribution

N(s)

and found that it is _a power law

N(s)

_«

I/s~.

The exponent T is

approximately

lA

(see Fig. 3).

We have also

investigated

the distribution of the _{seismic moment as} defined _{above and} found it to follow the _same power law _as

N(s).

We have not detected any

dependence

of the exponent

on the value of A in

(44)

_[17]. Also it does not _seem to matter if _we put Hi to be _constant

or random after rupture as

long

_as the initial value of _{H, is} random. The exponent ^for

N(s)

agrees with that found in [14] ^within

computational

_errors.

iooo ^' iooo

_~

100

~p

100

/~

-

~ ~G

10 1°

io ioo i io ioo

a)

s s

~~

Fig.3. (a)

Size distribution of earthquakes generated in the stationary state of the model for _a 100 X 100 lattice in the _case of time independent yield strength. 10000 earthquakes _are included to

obtain the statistics. The linear behavior of the log-log plot indicates _a Gutenberg-Richter law with

an exponent approximately I Al. The parameters used in the simulation _{are p}

= 1.0 X 10~~, _Hi

= 0.25,

x = 0.5 and _@u

= I-o- (b) Size distribution of earthquakes generated in the stationary state of the model for _a 100 _x 100 lattice in the _case of weakening and healing. 10000 earthquakes _are included to obtain the statistics. The linear behavior of the log-log plot indicates _a Gutenberg-llichter law with

an exponent approximately 1.38. The parameters used in the simulation _{are p}

= 1.0 _x 10~~, _@j

= 0.25,

x = 0.5 and _@u

= 1.0. The healing constant is A

= 10~~

Figure

4 shows _some

typical large

and _a medium size events. The pattern of

ruptured

sites associated with _a

single

event does not

depend

a great ^deal on the

healing law,

but is

largely

determined

by

the direction of stress enhancement after rupture which is

largest

_near the crack

tips.

The model which _we have described _so far has

only

_one time scale: the slow

geological

time scale which determines the accumulation of shear stress. This time scale is very

large.

In _a real

earthquake

_sequence,

however,

there appears ^to be _a second and _a smaller time scale.

Although

the main shock lasts

only

for about _a minute, the entire

earthquake

_sequence

combining foreshocks, mainshock,

and aftershocks _can last for _many weeks. The aftershock sequences are found _to

obey

^{Omori's law} [18]: ^the rate of aftershocks

following

the main shock

decays

as

I/tf

_{as a} function of time, with the exponent

(

close _{to one.} The most

popular explanation

of Omori's law is static

fatigue [19,

6], ^{that is the rock}

strength

is

time-dependent.

When the

applied

stress _a is below the instantaneous

breaking strength

but is above the _stress- corrosion limit Hoi ^rocks

generally

break with

probability

_per unit time proportional to e"', where _a is _a constant.

To illustrate this

general physical picture,

we have modified _our model to

incorporate

the

physics

of static

fatigue ill.

Consider _a

specific plaquet

at the position _ro in _our model. If the shear stress

a,(ro)

is above the instantaneous

breaking

threshold of the

plaquet H,(ro),

the

plaquet

will fracture _as described above.

However,

when the stress is below _H, but is still above Ho, the

plaquet

is set to break with

probability

_per unit time

given by e"("~°,)

Now, an

earthquake

_{sequence can} be defined _as follows. When all local stresses _are below the

respective

instantaneous

breaking

thresholds _as well _as the stress corrosion limit Ho,

nothing

will

happen:

this is in the

period

between

earthquake

_sequences, which _we call the _{rest state.}

100

. ,

100

,.

.

" ^°

,

80 80 _,

.

~

~

~~

~~

~',

~

~~

~~

/~(

~~

~~

~~

.

'

'

^'

,, ^{' ° '}

~~

"

'~£".

^~~

"(

,,., '"

0 0

0 20 40 60 80 loo 0 20 40 60 80 loo

~~~

,,~

_" ^~~~

80 ^" 80 ,

~

"

60

~

60 ^{" "}

/"

',

,. z

~~

',

~~

~

" ]~

20

~

~

20

~

~

"

$

0 0 ^"

0 20 40 60 80 loo 0 20 40 60 80 loo

FigA.

Some typical spatial patterns of simulated earthquakes.

100

~

o°

o 80

8~

o 60

y

^dP

40

,

~~

df

~

%

o

~o

0 20 40 60 80 00

x

Fig.5. Spatial

pattern ^ofa main

quake (full squares)

and aftershocks

(open circles).

The parameters used in the simulation _are the _{same as} those in figure 3b.

Whenever there is _a

place

where the stress is above the local instantaneous

breaking

threshold

or the _stress corrosion limit Ho, ruptures ^{in the} systems are

expected,

and _an

earthquake

sequence

begins.

The return of the system to the _rest state

signals

the end of this

earthquake

sequence. Each

earthquake

_{sequence can} contain _many

shocks, separated by periods

of _no

activity

with _some sites

remaining

to be fractured

by

static

fatigue.

The

largest

shock in _a sequence is defined to be the main

shock;

the shocks before the main shock _are foreshocks and the shocks after it _are aftershocks. As shown _{in our}

previous

_paper

ill,

this model will

obey

Omori's

law,

with _an exponent which is close to I. The

spatial

pattern associated with

a

typical

series of events is shown in

figure

5.

4. Discussion.

We have derived in detail _a novel discretization scheme _to represent ruptures in elastic media.

The method has the

advantage

of

being

fast and _easy to

implement,

and it is

quite

accurate

in _cases where _we have been able to compare with

analytical

results. The method _can

easily

incorporate

local disorder in

yield strength

and _can deal with

phenomenological

corrections associated with static

fatigue

and

healing.

We have illustrated the method _on two-dimensional square lattices with _open

boundary

conditions. The

boundary

^conditions _can

easily

be modified in the _case where the active

region

has _an

arbitrary shape,

but the _use of "non

open" boundary

conditions would

complicate

the calculation

by destroying

the translational invariance of the Green's functions. Similar difficulties _occur if _we attempt to deal with _a medium with _a

heterogeneous

elastic modulus. A modification of the model which would make it _more realistic without

causing

_a

great

deal of

complication,

is _to allow ruptures ⁱⁿ

arbitrary

directions and _we expect ^to present ^{results of such} simulations elsewhere. The

generalization

to three dimensions is also

quite straightfomard.

When

applied

to

earthquakes

the model

will, just

as several other models

reproduce

_some of the size and

temporal scaling

laws. What is _new is that there is also _a

possibility

of

studying spatial

patterns. Since the model is rather

flexible,

_we have concentrated _on

showing

how to

implement

it and

giving

_some

examples.

We therefore have

by

no means been able to

explore

the full parameter space.

Acknowledgments.

This work is

supported by

the Natural Sciences and

Engineering

Research Council of Canada.

References

[ii

Xu H-J., Bergersen B. and Chen K., J. Phys. A: Math. Gen. 25

(1992)

L1251.

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(North-

Holland, ^Amsterdam 1990).

[3] Knopoff L., Disorder and Fracture, J. C. Charmet Ed.

(Plenum,

New York,

1990).

[4] Dieterich J. H., J.Geophys. Res. 77 (1972) 3690.

[5] Scholz C. H., The mechanics of earthquakes and faulting

(Cambridge

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[6] ^{Atkinson B.} K., ^J. Geophys. Res. 89 (1984) 4077.

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1991).

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