2. Constitutive law for rock behaviors 3. Application to temporal seismicity patterns Yusuke Kawada & Hiroyuki Nagahama (DGES, Tohoku Univ.) mail to: [email protected] Temporal Power-laws on Preseismic Activation and Aftershock Decay Affected by Transient Behavior of Rocks Temporal Power-laws on Preseismic Activation and Aftershock Decay Affected by Transient Behavior of Rocks 4. Summary The 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 points We relate temporal 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 of irreversible thermodynamics [4] and time-scale invariance [5] . (with internal state valuables) We show that the temporal seismicity patterns are 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 t c - t (sec)] s = 0.20 log [Cumulative Benioff strain-release 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 ( ) t t s φ Ω Ω s − − = c c ( )( ). 2 3 1 2 − − = s s β Based on the fibre-bundle model [11] , and s are linked by β c : at occurrence of mainshock φ : constant Modified Omori's law [3] ( ) p c τ B τ d dN − + = Based on the continuum damage model [12] , and p are linked by β p. β 1 1 − = N : number of aſtershocks B : 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 14.0 1.5 2.0 2.5 3.0 3.5 log [ E S (MPa)] log [ (sec)] ξ = 0.15 β (1/ = 6.7) β = 0.07 β (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% = 0.13 β (1/ = 7.5) β = 0.03 β (1/ = 25.0) β Ref. T : 100°C Ref. : 1% ε (Kawada & Nagahama, 2004) Marble Analytical results of rock deformation [5] () () − = ′ = ≡ − RT Q C t ξ ξ ε g E ξ E ε σ β exp , S ξ : temperature reduced time () − ′ = RT Q σ C ε E ε ε g ε β β exp 1 1 Variation of exponent (stress exponent) [10] 1/ β Mechanism Material 1.3 Diffusion creep Anorthite 2.6 Dislocation creep Quartzite 3.0 Dunite 6.7 Halite 15.0 Primary creep Halite 27.0 Brittle failure Plagioclase 32.0 Brittle failure Granite 60.0 Brittle failure Sandstone Dislocation creep Dislocation creep () () ( ) ∫ − = ∞ λ d λ t λ D t E 0 exp () λ D : Distribution function of λ - - - - - - - - temporal fractral relaxation [9] . generates Structural fractal property ⇔ () β λ λ D − − ∝ 1 () β t t E − ∝ Lagrange equation for irreversible process [4] q : state valuables (generalized coodinates) [4] q F Γ dt dq ∂ ∂ − = regulated by [6] strain, damage parameter [6] accerelated plastic strain [7] [ ] F : free energy Γ : constant i j j ij j j ij Q q dt d b q a = + ∑ ∑ a : elastic coefficient Q : generalized force (external force) b : viscosity coefficient depending on the damage parameter [6] effect of hysteresis in the multiple step stress relaxation [8] ( ( ) ∫ ′ ′ ′ − = ξ ξ d ξ d ε d ξ ξ E ε d dq h a σ 0 e e ~ = General constitutive law [8] ⇔ ⇔ ⇔ () ( ) () ξ ε g E β ξ E ε d σ d β 1 ′ − = ≡ − () () ξ E ε g ~ 1 = is constitutive law includes the effect of damage q. Surface displacement [1] () () ′ = ≡ − ε g E E ε σ β S t t 5.0 4.0 3.0 2.0 1.0 0.0 2000 2001 2002 2003 Time [year] Cumulative displacement [cm] Regressive curve n = 3.5, A = 3.6×105 (MPa -n s -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( ) includes the 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 = 0.95 β log [Cumulative displacement (cm)] log [Time (day)] An earthquake (M L = 4.8) on Oct. 3, ‘72. San Juan Bautista ‘69 ‘70 ‘71 ‘72 ‘73 ‘74 ‘75 ‘76 ‘77 ‘78 ‘79 ‘80 Time [year] Cumulative displacement EQ (M L = 4.8) on Oct. 3, ‘72. 1cm Regressive curve n = 1 and U = 0 in Eq. (5) (Newtonian curve) San Juan Bautista (Wesson, 1987) (San Andreas Fault) Observed by creep meter ese 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 () t ε ε ε K t σ f f f , = = A fibre is subjected to the stress () ( ) () t n σ ν dt t dn f − = ( ) ρ σ σ ν σ ν = 0 f f f Brakedown rule is constrained by the Weibull distribution () 0 0 f σ t n n σ = e exponents of temporal seismicity patterns are constrained by fractal structure of crustal rocks. () ( ) ρ t t t e dt Ω d 2 1 c a − − ∝ ∝ e Benioff cumulative strain -release is formulated by η : fraction () () () ( ) ρ t t t n t e η t e 1 c f a − − ∝ − = Elastic energy released in the acoustic emission events is formulated by () 2 f f 2 1 ε K t e = Generalized Omori's law p c τ c p τ d dN N − + − = 1 1 1 T N T : total number of aſtershocks 1 + ∝ ρ σ ε () ( ) [ ] ρ t t ρ n t n 1 c 0 0 − = ν ree Eqs. yield two relations: Released rate of aſtershock energy is calucurated by the continuum damage model. rt e : total energy of aſtershock sequense () β ζ τ τ d τ de e − − + ∝ 1 1 r rt 1 1 , . . . , , , . .