Earthquake generation in different stress states

24

Physics of the Earth and Planetary Interiors, 49 (1987) 24—29 Elsevier Science Publishers B.V., Amsterdam — Printed in The Netherlands

Earthquake generation in different stress states

*

Roman Teisseyre Institute of Geophysics, Polish Academy of Sciences, Pasteura 3, Warsaw (Poland) (Received July 23, 1986; revision ‘accepted January 12, 1987)

Teisseyre, R., 1987. Earthquake generation in different stress states. Phys. Earth Planet. Inter., 49: 24—29. A method of numerical simulation of an earthquake sequence is presented based on the stress evolution equations and the new rebound theory. A zone of earthquake preparation is modelled by a medium filled with parallel antiplane cracks. Some statistical properties of the numerically simulated sequences are discussed; it follows that a complete representation of a cumulative energy diagram requires that energies released in the creep events should also be included.

1. Introduction

tion

An extension of the new earthquake rebound theory (Teisseyre, 1985a,b) is presented here and the method of numerical simulation of an earthquake sequence is outlined. We consider a medium with crack density a~( x1, x2, I) here parallel cracks extend in the x3 direction, and with the velocity field of a bulk crack motion V(x1, x2, I). The dislocation density a is defined by a num—

ber of dislocation lines crossing a unit surface and multiplied by a displacement (Burgers) vector L~u. This density is proportional to a stress concentration gradient dS/dx (Teisseyre, 1985). However, when referring to crack density one takes into account the equivalence theorem between a crack and dislocation array (Stroh, 1954). The number of dislocations forming a crack is proportional to 1a. A a localcrack stress motion concentration 5, hence cx S~ of bulk V refers to thea~motion defects as described by the density function, hence the viscous deformation rate is given by the rela*

Paper presented at the 16th International Conference on Mathematical Geophysics, Oosterbeek, The Netherlands, June 22—28, 1986.

0031-9201/87/$03.50

© 1987 Elsevier Science Publishers B.V.

= aV The shear field S(x1, x2,

t)

accounts for internal

stresses; this field undergoes (extension, an evolution process due to crack development grouping,

jstate. oimng). considerequations a kind of a visco-plastic TheWe constitutive expressing a relationship between antiplane crack density, and internal stresses lead and to the the equation describing crack flow motion evolution equation of stresses (Teisseyre, 1985a,b) (S

—

F )3

=

c

dS

(1) where we have taken a case of motion in the x~ direction and we have assumed that all functions depend on the variable x = Vt; C is a material constant. We also introduce here the friction stresses F, which are assumed to be velocity and velocity rate dependent. We will return to this question. A quantitative behaviour of the corresponding solutions has been discussed in previous papers. To solve it numerically, we add an additional condition for bulk crack velocity and stresses. We use here the energy balance criterion for rates; it —

25

takes the form

for quasi-static case, and dV

j a~i au2 +4_~__S2l =

—V(1

—

V2)~/2K2ac_

VFa

(2)

The left-hand side represents the rate of kinetic and strain energies, while the right-hand side describes terms related to a material flow. Here the first term is due to a quasi-static crack growth: an energy rate needed for the creation of a unit surface of crack— this expression has been derived by Kostrov et al. (1969) and it is here multiplied by crack density a~we shall take into account the difference between dislocation and crack density, the later being proportional to S~ dS/dx. The stress intensity factor K is assumed to be proportional to internal stress S. It can be shown that transition to a dynamic case is obtained when putting K K (1 I V 1)1/2 (Eshelby, 1969). A second term represents rate of friction work. We shall also note that in our system 8u/3t = —Vdu/dxcx —VS. We introduce here the non-dimensional quantities for velocity (in units of shear wave velocity), for stresses and friction = 1). After some steps and with new definition for constants (according to the non-dimensional units) we get the following .

—~

—

+(1 + V2)S]S/(CV2)

(4)

for a dynamic case. The friction is assumed to be velocity and velocity rate dependent, also some kind of barrier is introduced here in order to limit crack speed when approaching unity (shear wave velocity) cf. Fig. 1. —

2. Creep and earthquake sequence—rebound motion Now the system of the nonlinear equations (1) and (3) or (4) can be solved numerically. We get a kind of simulation procedure generating earthquake and creep events. At each computational step we divide the increments by I V I passing thus

(~

equations relation (2)for velocity gradient from the balance dx

=

[—ti

—

V2)’/2S_F+ (1 +

V

V2)S]S

/(CV2)

(3)

v
response toratel

~~O)

0 Fig. 1. Velocity and velocity rate dependent friction: schematic diagram.

Fig. 2. Extension of crack (left diagram) and crack joining in rebound motion V < 0 (right diagram)—a case of the in-plane shear cracks.

26

into a time domain. Here we take the regional I

stresses as the boundary conditions and we shall find a proper threshold condition for stress drop

2 H-O

S

0_________

I

H.0.4

I o

I

—

—.

I”

/

IH.08 20HQJ~ I

and energy release. Under the same stresses a flow velocity can according to eq. 1 reach both positive and negative values. In the rebound theory (Teisseyre, 1985b) aconsideration will inconservation fact separately describe a defect densitydensity of the opposite crack tips.law A rigorous including

10

~

—

~

—

H0 V

-I

0

Fig. 3. Stresses and stress drops (double lines) for different H values; stresses in relative (S/Sregionai) conventional scale (5. 5reg,onal = 1), velocity in shear wave velocity unit, time in conventional scale.

for a dislocation density (Teisseyre, 1985b) accounts for dislocations of one type (those having the same sign). A single crack is equivalent to two arrays of dislocations (Eshelby et al., 1951), each of different signs; they refer to different ends (tips) of the crack and move in opposite directions in a given stress field (crack extension). Thus we shall in fact use two different dislocation densities a and related to the given dislocation signs, hence also we should introduce bulk crack motion

fi

-I 4-

U

I‘4-

Fig. 4. Fragment of an earthquake sequence for 2% increase of regional stress field per unit time; stresses and friction in relative conventional scale, for stresses 5 . Sr 0gj0~ai= 1, for friction 10•~ = 1, velocities in shear wave velocity unit.

27

V~and ~ Our equations can be further simplified using relative velocity V = (aJ’~,+ /3V~)/(a+ $). For positive values of V we find that (for positively defined material constant) according to eq. 1 stresses increase. However, for negative values (V < 0) we observe a stress decrease, which relates to crack grouping and concentration; stress drop condition shall be inserted into the theory by additional conditions. We will return to this queslion. Taking this into account, we can ascribe to the positive values of a relative velocity a simple crack extension, while to the negative velocities, we ascribe a crack grouping and joining (Fig. 2—for a better demonstration we have taken here the in-plane shear cracks). The rebound motion

I

I I

2~,, a’ In as

/ I

It

I

domain corresponds to the negative velocities (Teisseyre, 1985). A cumulative rebound motion leads to fracture process accompanied by stress drop and energy release. In our theory it is assumed that a stress drop takes place when rebound motion overpasses a threshold velocity related to dynamic process or when rebound velocity reaches a maximum. The corresponding stress drop values are assumed to be proportional both to actual stress level and crack flow velocity = 115 I I (5) ~,

where 0 ~ H < 1. A coefficient H can be here evaluated by assuming that we look after the most effective earth-

50/

I

~,l

I

I

I

-I 0.2 I., •1~ ‘4-

Fig. 5. Fragment of an earthquake sequence for 5% increase of regional stress field per unit time, scale and units as in Figs. 3 and 4.

28

quake process. To define it, we shall first consider the case with H = 0—no stress drop. In this case a rebound domain (V < 0) describes slow energy release by the creep processes only (total creep rebound). We define the rebound domain by integral f~ I V Id t = d, where d represents a linear distance in which a creep process takes place. Analysing the stress and velocity fields, we found that d is the greatest for H = 0 or when H approaches 1, while for a certain value a rebound domain reaches a minimum dH. For our case it is H 0.4; in Fig. (3) the course of S and V for H = 0, H = 0.4 and H = 0.8 is presented. The above precise condition is in our case equivalent to a requirement that a creep time after an earthquake will be the shortest possible: f~,<~ dt = mm. Now we can define the energy release values. For H = 0 a total creep energy ETC would be proportional to an integral over a creep surface domain with rebound velocity V < 0 (in the in-

tegrand we take the quantity proportional to the energy release density rate aVS) ETC cx

f

I aVS I

ERC cx

f

I aVS I =

ETC

—

f

I

calm creep quake

)

double

•

quake

—

/

/

600

ERC

(8)

Thus we get the main independent parameters of a sequence. It shall be noted also that in some cases the stresses quickly dissipate falling down to a friction level; it is assumed that in such a situation

____

400

(7)

Hence an earthquake energy will equal to Eearlhquake

~I~Eearth+ Ecreep)

200

dt

V
500

500

(6)

This energy defines a creep energy release when no earthquake is produced. However, when assuming that a stress drop related to an earthquake process takes place, we get that for any H ~ 0 a remanent creep rebound energy ERC would be

—

bOO

dt

V
800

1000

time

Fig. 6. Sums of the earthquake and creep energies; energy and time in conventional scales.

29

only creep release takes place. Here in our model, the simulated sequences can be related rather to sequences of strong events; some additional conditions related to stress drops in a rebound domain might account for the effects of weaker events. We shall also note that to stabilize the numerical procedure, we need to introduce the additional friction which increases with time duration of a rebound process; such friction may be interpreted as related to heating along the fault plane.

ones observed in some regions to find the proper values of the parameters of the model presented. In Fig. 6 we present the cumulative energies released in the sequences simulated under regional load with increasing stresses by 2 and 5% per unit time (arbitrary scale). We have calculated not only the released energies in earthquakes but also in creep processes (by the appropriate integrals over the rebound motion domains). For a higher loading we can also observe the double events, while for a lower load rate the events occur more rarely.

3. Cumulative parameters of sequences in different stress states

References

A sequence is now defmed with events characterized by their moment of occurrence, creep energy, seismic energy and stress drop. Some examples of the fragments of such simulated sequences are shown in Figs. 4 and 5; the sequences discussed here are related to the same material parameters and the same initial conditions, but differ in the boundary conditions: one sequence is related to the regional stresses which increase at the 2% rate per umt time while another at the 5% rate. A graph of cumulative energies or stress drops can be compared with those similar

.

Eshelby, J.D., 1969, The elastic field of a crack extending non-uniformly under general anti-plane loading. J. Mech. Phys. Solids, 17: 177. Eshelby, J.D., Frank, F.C. and Nabarro, F.R.N., 1951. The equilibrium of linear arrays of dislocations. Philos. Mag., 42: 351—364. Kostrov, B.V., Nikitin, LV. and Flitman, L.M., 1969. Mechanics of brittle fracture. Izv. Akad. Nauk SSSR Mekh. Tverdogo Tela, 3: 112—125. Stroh, AN., 1954. The formation of cracks as a result of plastic flow. Proc. R. Soc. London, A 223: 404—414. Teisseyre, R., 1985a, New earthquake rebound theory. Phys. Earth Planet. Inter., 39: 1—4. Teisseyre, R., 1985b. Creep flow and earthquake rebound: system of internal stress evolution. Acta Geophys. Pol., 33: 11—23.