1. Laurent G. J. Montési Maria T. Zuber 6-30-99 ASME, 1999 The importance of localization for the development of large-scale structures in the Earth’s crust MIT

2. Central Indian Ocean  Regularly spaced faults Wavelength ~ 7 km No apparent decollement or material transition. Multichannel seismic reflection, Jestin, 1994 10 km

3. Origin of the spacing of localized zones  Buckling of viscous and/or elastic media. Biot 1957, 1961, Fletcher and hallet, 1983, Zuber et al. 1986.  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

4. Thrust fault, Alaska

5. Definition of the effective rheology  General rheology:  Parameterize using  0 :  Define the rheological derivative.   0 -potential:

6. Stability of the rheological law  Differential of the potential:  Imposed perturbation:  Effective stress exponent: Localizing condition: n e negative

7. Rheology trajectories

8. 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.

9. Rate- and state-dependent friction  Constitutive law in steady-state  Effective stress exponent  Possible stabilization by elastic coupling  Transient effects delay instability

10. 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.

11. Poiseuille flow

12. Analytical model

13. Perturbation analysis  Basic deformation: uniform shortening and thickening.  Solution of Stokes flow, incompressible uniform fluid layer.   real for n<0

14. 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.

15. Growth spectrum:  Two styles of deformation.  Two different wavelengths.

16. Growth rate map  New branches at negative n.  Matched by resonance between modes at different a Localization at specific wavelength

17. 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.

18. Initial model  Aspect ratios: grid: 5x2 elements:1x1 to 1x4 localizing layer: 10x1   Shortening rate: 2%/Ma   Time step: 10000 years   Viscosity contrast: ~0.1   c=0.1

19. 8% shortening 32% shortening

20. Wavelength evolution

21. 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, 1995.  Hydrostatic pressure  Error-function geotherm.

22. 14.3% shortening 5.4% shortening

23. Venusian Ridge Belts  Ridges spacing: 1-2 km.  Longer wavelength (300 km). Magellan radar mosaic

24. 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.