2. Constitutive law for rock behaviors

3. Application to temporal seismicity patterns

Yusuke Kawada & Hiroyuki Nagahama (DGES, Tohoku Univ.)mail to: y-kawada@mail.tains.tohoku.ac.jp

Temporal Power-lawson Preseismic Activation and Aftershock DecayAffected by Transient Behavior of Rocks

Temporal Power-lawson Preseismic Activation and Aftershock DecayAffected by Transient Behavior of Rocks

4. SummaryThe temporal seismicity patterns on the surface displacement and prior or

subsequent to the mainshocks can be recognized by the time-scale invariance of the transient behavior of rocks. These patterns are analyzed by the constitutive law derived from the irreversible thermodynamics which is linked to the fibre-bundle model or continuum damage model. The change in exponents of the temporal power-laws on the cumulative Benioff strain-release and modified Omori's law may be regulated by the fractal structure of crustal rocks in response to the different deformation mechanisms.

1. Introduction and pointsWe relatetemporal seismicity patterns and constitutive law of rock behaviors

・surface displacement[1]

・preseismic activation[2]

・aftershock decay[3]( ) ・transient and steady-state creep・stress relaxation・brittle behavior and failure

( )in terms ofirreversible thermodynamics[4] and time-scale invariance[5].

(with internal state valuables)

We show that

the temporal seismicity patternsare regulated by

the fractal property of crustal rocks.

5. Appendix

References: [1] Freed, A.M., Bürgmann, R. (2004) Nature 430, 548; Wesson, R.L. (1987) Tectonophysics 144, 215. [2] Bowman, D.D. et al., (1998) J. Geophys. Res. 103B, 24359; Bufe, C.G., Varnes, D.J. (1993) J. Geophys. Res. 90B, 12575. [3] Utsu, T. (1961) Geophys. Mag. 30, 521. [4] Biot, M.A. (1954) J. Appl. Phys. 25, 1385; Schapery, R.A. (1964) J. Appl. Phys. 35, 1451-1465. [5] Kawada, Y., Nagahama, H. (2004) Terra Nova 16, 128. [6] Lyakhovsky, V. et. al., (1993) Tectonophysics 226, 187. [7] Lemaitre, J. (1985) Trans. ASME, J. Appl. Mech. 107, 83. [8] Schapery, R.A. (1969) Polym. Eng. Sci. 9, 295. [9] Nagahama, H. (1994) In: Fractal and Dynamical Systems in Geosciences. Springer, Berlin, p. 121. [10] Nakamura, N., Nagahama, H. (1999) In: Atmospheric and Ionospheric Electromagnetic Phenomena Associated with Earthquakes. TERRAPUB, Tokyo, p. 307. [11] Turcotte, D.L. et. al., (2003) Geophys. J. Int. 152, 718. [12] Nanjo, K.Z. et. al., (2005) J. Geophys. Res. 110, B07403.

4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5

log [Remaining time tc - t (sec)]

s = 0.20

log

[Cum

ulat

ive

Beni

off

stra

in-r

elea

se

c -

(J

) ]Ω

Ω s = 0.16

s = 0.0070

s = 0.0025

6.0

6.1

6.2

6.3

6.4

6.5

6.6

(b) North-eastern Caribbean Sea

(a) Himachal Himalaya

Cumulative Benioff strain-release[2] Ω

( )ttsφΩΩ s−−= cc

( ) ( ).2312 −−= ssβ

Based on the fibre-bundle model[11], and s are linked byβ

c : at occurrence of mainshockφ : constant

Modified Omori's law[3]

( ) pcτBτd

dN −+= Based on the continuum damage model[12], and p are linked byβ

p.β 11−=N : number of aftershocksB : constant

τ : occurrence time of mainshock

10%

7%

5%

3%

1%

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.01.5

2.0

2.5

3.0

3.5

log

[ E S

(MPa

)]

log [ (sec)]ξ

= 0.15β(1/ = 6.7)β

= 0.07β (1/ = 15.0)β

Ref. T : 100°C Ref. : 1% ε

(Shimamoto, 1987)

Halite

2.5

3.0

3.5

4.0

4.5

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5

log

[ E S

(MPa

)]

log [ (sec)]ξ

10%

8%

6%

4%

2%

= 0.13β(1/ = 7.5)β

= 0.03β (1/ = 25.0)β

Ref. T : 100°C Ref. : 1% ε

(Kawada & Nagahama, 2004)

Marble

Analytical results of rock deformation[5]

( ) ( )

−=

′=≡ −

RTQ

Ctξξ

εgEξE

εσ β exp,S

ξ : temperature reduced time

( )

−

′=

RTQσ

Cε

Eεεgε ββ

exp11

Variation of exponent (stress exponent)[10]

1/β Mechanism Material1.3 Diffusion creep Anorthite2.6 Dislocation creep Quartzite3.0 Dunite6.7 Halite15.0 Primary creep Halite27.0 Brittle failure Plagioclase32.0 Brittle failure Granite60.0 Brittle failure Sandstone

Dislocation creepDislocation creep

( ) ( ) ( )∫ −=∞

λ dλtλDtE

0 exp ( )λD : Distribution function of λ

- - - -

- - - -

temporal fractral relaxation[9].

generatesStructural fractal property

⇔

( ) βλλD −−∝ 1

( ) βttE −∝

Lagrange equation for irreversible process[4]

q : state valuables (generalized coodinates)[4]

qFΓ

dtdq

∂∂

−=regulated by[6]

strain, damage parameter[6] accerelated plastic strain[7][ ]

F : free energyΓ : constant

ij

jijj

jij Qqdtdbqa =+∑∑

a : elastic coefficient

Q : generalized force (external force)

b : viscosity coefficient

①

・depending on the damage parameter[6]

・effect of hysteresis in the multiple step stress relaxation[8](

( )∫ ′′

′−=ξ

ξdξdεdξξE

εddqhaσ

0 ee~

=

General constitutive law[8]②

③

⇔⇔⇔

( ) ( )( ) ξεg

EβξEεdσd β1 ′−

=≡ −

( ) ( )ξEεg

~1=

This constitutive law includesthe effect of damage q.

Surface displacement[1]

( ) ( )′

=≡ −

εgEE

εσ β

S t t

5.0

4.0

3.0

2.0

1.0

0.02000 2001 2002 2003

Time [year]

Cu

mu

lati

ve d

isp

lace

men

t [c

m]

Regressive curve

n = 3.5, A = 3.6×105 (MPa-ns-1),and U = 480 (KJ mol-1) in Eq. (5)

(Freed & Bürgmann, 1993)

Mojava desert (South California) after 1999 Hector Mine EQ

Observed by GPS

g( ) includesthe effect of hysteresis.

ε

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

= 0.95β

log

[Cu

mu

lati

ve

dis

pla

cem

ent

(cm

)]

log [Time (day)]

An earthquake (ML = 4.8)on Oct. 3, ‘72.

San Juan Bautista

‘69 ‘70 ‘71 ‘72 ‘73 ‘74 ‘75 ‘76 ‘77 ‘78 ‘79 ‘80

Time [year]

Cu

mu

lati

ve d

isp

lace

men

t

EQ (ML = 4.8)on Oct. 3, ‘72.

1cm

Regressive curven = 1 and U = 0 in Eq. (5)(Newtonian curve)

San Juan Bautista

(Wesson, 1987)(San Andreas Fault)

Observed by creep meter

These time series due to major earthquakes with small events can be recognized by the temporal fractal property on our constitutive law of rocks.

Fibre-bundle model

( )tεεεKtσ f

ff , ==

A fibre is subjected to the stress

( ) ( ) ( )tnσνdt

tdnf−=

( )ρ

σσνσν

=

0

fff

Brakedown rule is constrained by the Weibull distribution

( ) 00

f σtn

nσ =

The exponents of temporal seismicity patterns are constrained by fractal structure of crustal rocks.

( ) ( ) ρtttedtΩd

21

ca−−∝∝

The Benioff cumulative strain-release is formulated by

η : fraction

( ) ( ) ( )

( ) ρtt

tnteηte1

c

fa −−∝

−=

Elastic energy released inthe acoustic emission events is formulated by

( ) 2 ff

21 εKte =

Generalized Omori's lawp

cτ

cp

τddN

N

−

+

−= 111

T

NT : total number of aftershocks1+∝ ρσε

( ) ( )[ ] ρttρntn 1c00 −= ν

Three Eqs. yield two relations:

Released rate of aftershock energyis calucurated by the continuum damage model.

rte : total energy of aftershock sequense

( ) β

ζτ

τdτde

e−

−

+∝1

1 r

rt

11,

.

.

.

,

,

,

.

.