Laurent G. J. MontésiLaurent G. J. Montési

Maria T. ZuberMaria T. Zuber

6-30-996-30-99ASME, 1999ASME, 1999

The importance of localization The importance of localization for the development of large-for the development of large-scale structures in the Earth’s scale structures in the Earth’s crustcrust

MITMIT

Central Indian OceanCentral Indian Ocean

Regularly spaced faultsWavelength ~ 7 kmNo apparent decollement or material transition.

Multichannel seismic

reflection,

Jestin, 1994

10 km

Origin of the spacing of localized Origin of the spacing of localized zoneszones

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

Thrust fault, AlaskaThrust fault, Alaska

Definition of the effective rheologyDefinition of the effective rheology

General rheology:

Parameterize using 0:

Define the rheological derivative.

0-potential:

i

00

dd

dd

ii

00

dð

ðd

0

0 ð

ð ;d

PP

Final

Initial

Stability of the rheological lawStability of the rheological law

Differential of the potential:

Imposed perturbation:

Effective stress exponent:

enP 11dddd 000

0d

0

0

ð

ð1

en

Localizing condition: Localizing condition: nnee negativenegative

Rheology trajectoriesRheology trajectories

0

UnstableUnstable

Direct R

esponse

System Response

Some localization mechanismsSome localization mechanismsBrittle domain:

Friction velocity weakeningCohesion 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 Rate- and state-dependent frictionfriction

Constitutive law in steady-state

Effective stress exponent

Possible stabilization by elastic coupling

Transient effects delay instability

00 ln VVba

bane

Negative stress exponent?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 flowPoiseuille flowN

on-lo

caliz

ing

n = 1n = 5n = 1030

Scaled velocity

Loca

lizin

g

n = -5

Analytical modelAnalytical model

x

2

Localizing1, n1=-10

Ductile2=1 n2=3

1

z

Perturbation analysisPerturbation analysis

Basic deformation: uniform shortening and thickening.Solution of Stokes flow, incompressible uniform fluid layer.

real for n<0

zxikt ji

jj exp,0

0 1 12

22 4

n

012

2 42

24

4

4

j

jjk

dz

dk

ndz

d

0 0.5 1 1.5 2 2.5-1.5

-1

-0.5

0

0.5

1

1.5

5

5

-5 -5

2.5

2.5

-2.5

-2.5

1.6667

1.6667

1.25

1.25

-1.25 -1.25Inf

real(a)

imag( a)

Boundary conditionsBoundary 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.

j zz

ji

ixdt

d

tq zz 1exp0

Growth spectrum:Growth spectrum:

Two styles of deformation.Two different wavelengths.

Growth rate mapGrowth rate map

New branches at negative n.

Matched by resonance between modes at different a

Localization at Localization at specific specific

wavelengthwavelength

Finite Element MethodFinite Element Method

Layer, from Neumann and Zuber 1995Neumann and Zuber 1995

Retain the weakest of ductile and brittle strength.Ductile rheology:

Brittle rheology:

Lagrangian grid.

Constant shortening velocity.

Initial convergence/localization steps.

0II0 ln1 cbb

nd A 10II

Initial modelInitial model

0II10log 0

Aspect ratios:grid: 5x2elements:1x1 to 1x4localizing layer: 10x1

Shortening rate: 2%/MaTime step: 10000 yearsViscosity contrast: ~0.1c=0.1

0II10log 0

8% shortening

32% shortening

Wavelength evolutionWavelength evolution

Model of the earth’s crustModel 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 pressureError-function geotherm.

0II10log 0

0II10log 0

14.3% shortening

5.4% shortening

Venusian Venusian Ridge BeltsRidge Belts

Ridges spacing: 1-2 km.

Longer wavelength (300 km).

Magellan radar mosaic

ConclusionsConclusions

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.