Inherent Mechanism Determining

1
Inherent Mechanism Determining Scaling
Properties of Fault Constitutive Laws
Mitsuhiro Matsuura 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
Matsuura et al, 1992) - Rate- and
State-dependent law (e.g., Dieterich, 1979
Ruina, 1983) - Slip- and time-dependent law
(Aochi Matsuura, 1999, 2002) -
Scale-dependence of the critical weakening
displacement Dc 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
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Basic Equations Governing Earthquake Generation
Cycles
Slip Response Function
Shear stress change due to slip perturbation
Boundary conditions to be satisfied
Fault Constitutive Law
Change in fault constitutive relation with slip
and time
Total slip at a plate interface
Relative Plate Motion
4
Energy Balance for Spontaneous Rupture Growth
y
x
5
Slip-weakening Constitutive Law
Characteristic weakening displacement
Upper fractal limit
(b) Fractality of rock surfaces Power, et al.,
1987
(a) Observed constitutive relation Ohnaka, et
al., 1987
(c) Change in surface topography with fault slip
Matsuura, et al., 1992
(d) Theoretical constitutive relation Matsuura,
et al., 1992
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Quasi-static Rupture Nucleation Process Governed
by the Slip-weakening Constitutive Law
Asperity
3D plot of fault constitutive relations
Matsuura, et al., 1992 t Shear strength. w
Fault slip. x Distance along the fault.
Quasi-static shear stress (a) and fault slip (b)
changes with time Matsuura, et al., 1992
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Transition from Quasi-static Nucleation to
Dynamic Rupture
From observation and simulation
Fundamental scaling law
Shear stress change during dynamic rupture of an
asperity and the subsequent major event
Shibazaki and Matsuura, 1992
Change in fault slip (thick line) and slip
velocity (thin line) with time Shibazaki and
Matsuura, 1992
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The Entire Earthquake Generation Process
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Restoration of Fault Strength Log t-healing
during stationary contact and slip-velocity
weakening in steady-state slip
(b) Evolution of surface topography during
stationary contact Aochi and Matsuura, 2002
Characteristic healing time
(a) Change in fault strength with time in
stationary contact Dieterich, 1972
(c) Slip-velocity dependence of fault strength
in steady-state slip Dieterich, 1978
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Slip- and Time-dependent Fault Constitutive
Law Aochi and Matsuura, 1999, 2002
Definition of fault strength and the evolution
equation of surface topography
Physical quantities and parameters
Inherent Mechanisms - Slip weakening due to
abrasion of fractal rock surfaces - Strength
restoration due to adhesion and adhesive ware
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Constitutive Properties of the Slip- and
Time-Dependent Law
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Evolution Equation of the State Variable in a
Rate- and State-Dependent Law (NielsenI et al.,
2000)
The evolution equation of the slip- and
time-dependent law
For surface asperities with a characteristic
wavelength ( )
with
Characteristic displacement for slip-weakening
Characteristic time for healing
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Simulation of Complete Earthquake Generation
Cycles
Shear Stress Fault Slip
Shear Stress Slip Deficits
(b) Initial stress distribution
0 Shear Stress (MPa) 3
0 Slip Deficits (m) 2
(c) Dynamic rupture propagation
(a) Quasi-static stress accumulation
Hashimoto, Fukuyama Matsuura
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Evolution of Fault Constitutive Relation During
One Earthquake Cycle
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.
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
Matsuura, 2002.
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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.
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• Conclusions
• Fault constitutive laws play the role of an
  interface between microscopic processes in fault
  zones and macroscopic processes of a fault system.

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