1. Inherent Mechanism Determining Scaling Properties of Fault Constitutive Laws Mitsuhiro Matsu’ura Department of Earth and Planetary Science Graduate School of Science The University of Tokyo

2. Progress in the Physics of Earthquake Generation in 1990s ■ Introduction of Laboratory-based Fault Constitutive Laws as a Basic Equation Governing Earthquake Rupture - Slip-weakening law (e.g., Ohnaka et al., 1987; Matsu’ura et al, 1992) - Rate- and State-dependent law (e.g., Dieterich, 1979; Ruina, 1983) - Slip- and time-dependent law (Aochi & Matsu’ura, 1999, 2002) - Scale-dependence of the critical weakening displacement D c ■ Quantitative Description of Tectonic Loading Driven by Plate Motion - Viscous drag at the base of the lithosphere (base-loading) - Dislocation pile-ups at the edge of a locked portion (edge-loading) - Mathematical formulation of elastic/viscoelastic slip-response functions

3. Basic Equations Governing Earthquake Generation Cycles Fault Constitutive Law Slip Response Function Relative Plate Motion Total slip at a plate interface Shear stress change due to slip perturbation Change in fault constitutive relation with slip and time Boundary conditions to be satisfied

4. Energy Balance for Spontaneous Rupture Growth y x

5. Slip-weakening Constitutive Law (a) Observed constitutive relation [Ohnaka, et al., 1987] (b) Fractality of rock surfaces [Power, et al., 1987] (d) Theoretical constitutive relation [Matsu’ura, et al., 1992] (c) Change in surface topography with fault slip [Matsu’ura, et al., 1992] Dc ≈ : Characteristic weakening displacement Upper fractal limit

6. Quasi-static Rupture Nucleation Process Governed by the Slip-weakening Constitutive Law 3D plot of fault constitutive relations [Matsu’ura, et al., 1992]  :  Shear strength. w: Fault slip. x: Distance along the fault. Quasi-static shear stress (a) and fault slip (b) changes with time [Matsu’ura, et al., 1992] Asperity

7. Transition from Quasi-static Nucleation to Dynamic Rupture Shear stress change during dynamic rupture of an asperity and the subsequent major event [Shibazaki and Matsu’ura, 1992] Change in fault slip (thick line) and slip velocity (thin line) with time [Shibazaki and Matsu’ura, 1992] From observation and simulation Fundamental scaling law

8. The Entire Earthquake Generation Process

9. Restoration of Fault Strength Log t-healing during stationary contact and slip-velocity weakening in steady-state slip (a) Change in fault strength with time in stationary contact [Dieterich, 1972] (b) Evolution of surface topography during stationary contact [Aochi and Matsu’ura, 2002] (c) Slip-velocity dependence of fault strength in steady-state slip [Dieterich, 1978] Characteristic healing time:

10. Slip- and Time-dependent Fault Constitutive Law [Aochi and Matsu’ura, 1999, 2002] Inherent Mechanisms: - Slip weakening due to abrasion of fractal rock surfaces - Strength restoration due to adhesion and adhesive ware Physical quantities and parameters Definition of fault strength and the evolution equation of surface topography

11. Constitutive Properties of the Slip- and Time-Dependent Law The case of high-speed slip The case of stationary contact The case of steady-state slip Characteristic weakening displacement: Characteristic healing time: --> Slip weakening --> Log t healing --> Slip-velocity weakening

12. Evolution Equation of the State Variable in a Rate- and State-Dependent Law (NielsenI et al., 2000) For surface asperities with a characteristic wavelength ( ) : ; Characteristic time for healing The evolution equation of the slip- and time-dependent law: ; Characteristic displacement for slip-weakening with

13. Shear Stress Fault Slip Simulation of Complete Earthquake Generation Cycles (c) Dynamic rupture propagation Hashimoto, Fukuyama & Matsu’ura (b) Initial stress distribution 0 Shear Stress (MPa) 3 0 Slip Deficits (m) 2 Shear Stress Slip Deficits (a) Quasi-static stress accumulation

14. Evolution of Fault Constitutive Relation During One Earthquake Cycle Change in constitutive relation with time after a large earthquake: rapid restoration of peak strength and gradual increase of critical weakening displacement Dc [Hashimoto and Matsu’ura, 2002]. The critical weakening displacement Dc gradually increases with contact time t. The gradual increase of Dc with contact time t can be attributed to the gradual recovery of larger-scale fractal structure of damaged fault through adhesion of surface asperities in direct contact.

15. A Realistic Image of Fault Strength Restoration after the Occurrence of a Large Earthquake (b) A schematic diagram showing restoration of fault constitutive properties after a large event. (a) An image of the heterogeneous fault with a hierarchic fractal structure in Dc. Scale dependence of healing time: Scale dependence of Dc:

16. Macroscopic viewpoints: - Functional relation among shear strength, fault slip, and contact time - Basic equation governing earthquake rupture / Physics - Boundary condition in continuum mechanics / Mathematics Microscopic viewpoints: - Energy balance equation for a fault zone with fractal internal structure - Mechanical energy dissipation in fault zones / Slip-weakening - Restoration of fractal structure in fault zones / Strengthening in contact - Integration of microscopic physicochemical processes in fault zones Conclusions Fault constitutive laws play the role of an interface between microscopic processes in fault zones and macroscopic processes of a fault system.