Laurent G. J. Montési Maria T. Zuber ASME, 1999 The importance of localization for the development of large-scale structures in the Earth’s crust MIT
Central Indian Ocean Regularly spaced faults Wavelength ~ 7 km No apparent decollement or material transition. Multichannel seismic reflection, Jestin, km
Origin of the spacing of localized zones Buckling of viscous and/or elastic media. Biot 1957, 1961, Fletcher and hallet, 1983, Zuber et al How can one treat the localization of deformation? Define faults a-priori. Apply yield criterion a-posteriori. Slip-line fields. Define rheology during localization. Effective stress exponent. Adapt buckling theory. Analytical and numerical analysis
Thrust fault, Alaska
Definition of the effective rheology General rheology: Parameterize using 0 : Define the rheological derivative. 0 -potential:
Stability of the rheological law Differential of the potential: Imposed perturbation: Effective stress exponent: Localizing condition: n e negative
Rheology trajectories
Some localization mechanisms Brittle domain: Friction velocity weakening Cohesion loss. Pressure dilatancy. Non-associative plasticity. Ductile domain: Adiabatic shear localization. Conductive equilibrium. Grain-size sensitivity. Other possible feedback mechanisms: Phase transformation. Melt weakening.
Rate- and state-dependent friction Constitutive law in steady-state Effective stress exponent Possible stabilization by elastic coupling Transient effects delay instability
Negative stress exponent? Non-linearity of the rheology. Plastic behavior at n , or 1/n 0. Weakening of the active region: weak faults and plate tectonics. Effective viscosity for flow perturbation during buckling: /n.
Poiseuille flow
Analytical model
Perturbation analysis Basic deformation: uniform shortening and thickening. Solution of Stokes flow, incompressible uniform fluid layer. real for n<0
Boundary conditions Uniform layer over a non-localizing half-space. Match stresses and velocity across interfaces. Resolve evolution of interface perturbations. Select fastest growing mode. Construct growth spectrum.
Growth spectrum: Two styles of deformation. Two different wavelengths.
Growth rate map New branches at negative n. Matched by resonance between modes at different a Localization at specific wavelength
Finite Element Method Neumann and Zuber 1995 Layer, from Neumann and Zuber 1995 Retain the weakest of ductile and brittle strength. Ductile rheology: Brittle rheology: Lagrangian grid. Constant shortening velocity. Initial convergence/localization steps.
Initial model Aspect ratios: grid: 5x2 elements:1x1 to 1x4 localizing layer: 10x1 Shortening rate: 2%/Ma Time step: years Viscosity contrast: ~0.1 c=0.1
8% shortening 32% shortening
Wavelength evolution
Model of the earth’s crust Pressure-dependent frictional resistance (Byerlee’s law) with weakening. Temperature-dependent power law creep. Quartzite without melt, Gleason and Tullis, Hydrostatic pressure Error-function geotherm.
14.3% shortening 5.4% shortening
Venusian Ridge Belts Ridges spacing: 1-2 km. Longer wavelength (300 km). Magellan radar mosaic
Conclusions Localization can be modeled using an effective rheology with negative stress exponent. That approximation allows the theory of folding/buckling to be adapted to include localization. Model faults grow at a different wavelength from folding. At finite strain, the fault spacing is preserved, not the folding wavelength.