Basic concepts for Localization of...

Introduction lecture notes of Yehuda Ben-Zion, Localization seminar Fall 2009

Basic concepts for Localization of deformationBasic concepts for Localization of deformation

•Weakening vs. strengthening rheologies (P, T, porosity, fluids, grain size) –positive vs. negative feedbacks

•Stress vs. displacement/velocity boundary conditions - unstable/stable processes

•Effects of healing for displacement/velocity boundary conditions - ratio of healing/loading timescales

•Effects of heterogeneities (can suppress localization)

•Effects of inherited structures: bimaterial interfaces, weak zones, .....

•Effective behavior on different scales: single microcrack, macroscopic shear crack, cataclastic flow, fault zone, fault system, plate boundary, ……


•Weakening vs. strengthening rheologies (P, T, porosity, fluids, grain size) –positive vs. negative feedback mechanisms


Deformation band (Shear band)

Deformation zone

Goblin Valley, Utah


The Punchbowl fault: 44 km of slip (Chester and Chester, 1998)


•Stress vs. displacement/velocity boundary conditions - unstable/stable processes


Stability

Constant stress loading lead to dynamic instability once a critical crack length Lc has been archived (perhaps by creep).

Griffith experiments

In contrast, under constant displacement/velocity, failure is stable

Stability is NOT a material property but the response of the entire system (failure zone plus loading environment). Stability is determined by comparing rate of strength change in failure area with rate of loading reduction during failure (zero in a, finite in b). The latter is referred to as the “stiffness of the system”.

aa

aa

Constant displacement loading

Constant stress

Constant stress

unstable

stable

RK ∞∝τ


•Effects of healing for displacement/velocity boundary conditions - ratio of healing/loading timescales


We fix all the large scale parameters (e.g., dimensions, background elastic properties, viscosity) using data associated with the San Andreas fault.

The evolving results depend on the ratio of time scale for damage healing τH to time scale for tectonic loading τL

Coupled evolution of earthquakes and faults

Evolving ElasticUpper Crust

Viscoelastic Lower Crust

ViscoelasticMantle

(half space)

ν = 0.25

Distributed Damage(fault zones)

100 km

H = 15 km

Loading bydistributedsteady mantlemotion

h = 20 km


slow effective healing fast effective healing

α

Characteristic earthquakes Power-law statistics


•Effects of inherited structures: bimaterial interfaces, weak zones, .....


Weertman (1980): 2D analytical solution for steady state mode II slip pulse on a bimaterial interface governed by Coulomb friction.

In-plane slip: Moving coordinate system: ξ = x – ctDislocation density: B(ξ) = – dδ/dξ

δ(x,t)= u(x,y= 0+,t)−u(x,y= 0−,t)

The shear and normal stress on the interface are

u(x,y = 0+ ,t)u(x,y = 0− ,t)

•In a homogeneous solid μ* = 0; there is no coupling between slip and σσ.

•For subsonicsubsonic rupture on a bimaterial interface in the direction of motion ofthe compliant solid, μ*> 0 and σσ drops dynamicallydrops dynamically (producing local dilation).

•In the opposite direction, μ*< 0 and σσ increases dynamically increases dynamically (local compression).

••Adams (1995): The bimaterial effects Adams (1995): The bimaterial effects increase with propagation distance!increase with propagation distance!

compliantcompliant

stiffstiff)(),(*)( ξβμσξσ Bc Δ−= ∞

'')'(),()( ξ

ξξξ

πβμτξτ dBc

∫∞

∞−

∞

−Δ

+=


Wrinkle-like rupture on a bimaterial interface

Andrews and Ben-Zion, 1997; Ben-Zion and Andrews, 1998; Cochard and Rice, 2000; Ben-Zion, 2001; Ben-Zion and Huang, 2002, Shi and Ben-Zion, 2006; Ampuero and Ben-Zion, 2008; Brietzke et al., 2009; Dalguer and Day, 2009; …

compliant

stiff


Characteristic features of wrinkle-like rupture pulse:

1) strong correlation between variations of normal stress and slip

2) strongly asymmetric motion across the fault

3) preferred direction of rupture propagation

4) self-sharpening with propagation distance


Rupture migration in tri-material structure with multiple available rupture planes (Brietzke and Ben-Zion, 2006)

Parameter space study for different

•Nucleation locations

•Fault separation

•Initial shear stress

•Velocity contrasts


Example results

Coulomb friction Regularized Prakash-Clifton friction

faul

t num

ber

max. of slip−velocity on fault [m/s]

distance [m]−500 0 500

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What happens if we add additional ingredients?

•Stress heterogeneities (Ben-Zion & Andrews, 1998; Andrews & Harris, 2005)

•Low-velocity fault zone layer (Harris & Day, 1997; Ben-Zion & Huang 2002; Brietzke & Ben-Zion, 2006)

•Viscosity in the bulk (Harris and Day, 1997; Brietzke and Ben-Zion, 2006)

•Prakash-Clifton friction (Cochard & Rice, 2000; Ben-Zion & Huang, 2002)

•Contrast of permeability structure (Rudnicki & Rice, 06; Dunham & Rice, 08)

•Slip-weakening friction (Harris & Day, 97; Shi & Ben-Zion, 06; Rubin & Ampuero, 2007; Brietzke et al., 2007, 2008 )

•Creation of off-fault damage (Ben-Zion and Shi, 2005; Duan, 2008)

•Multiple possible rupture plans (Brietzke and Ben-Zion, 2006)

•Velocity-weakening friction (Ampuero and Ben-Zion, 2008)

•3D effects (Brietzke et al., 2007, 2009; Dalguer & Day, 2009)

There is a diversity of phenomena. However, the results show collectively the results show collectively that ruptures evolve that ruptures evolve for broad ranges of realistic conditionsfor broad ranges of realistic conditions to slip pulses in to slip pulses in the preferred direction.the preferred direction.


•Effective behavior on different scales: single microcrack, macroscopic shear crack, cataclastic flow, fault zone, fault system, plate boundary, ……


Con

tinuu

m M

echa

nics

Gra

nula

r M

echa

nics

stat

istic

al

mec

hani

cs

Frameworks for studying brittle deformationσ1−σ3

σ1

σ3ε

K, GFracture mechanics

Rock strength

Friction studies

Damage rheology


When material is stressed it deforms

Faulting angle typicallyaround 30 degrees!

Rock strength experimentsRock strength experiments

BC are veryimportant!


ductileductilebrittlebrittle

Increasing pressure leads to ductility

brittlebrittleductileductile

Increasing temperature leads to ductility

Brittle-ductile: stress-strain curves with permanent inelastic brittle and ductile deformation


Stress-strain and AE locations for Westerly granite (Lockner et al., 1992)


Lockner’s animations

Acoustic emission in fracturing experiments with Westerly granite and Berea sandstone


x0

t Gc t + ΔtX0 + Δx

•Surface area•Radiation•Heat (friction)•Plastic strain and damage

•Surface area•Kinetic energy (radiation)•Plastic strain and damage

•Surface area

Dynamic shearDynamic tensileQuasi-static tensile

Energy sinks:

•Fracture, cracking:Deformation Mechanisms

General definition: localized deformation converting elastic strain energy to surface energy Gc associated with cohesion (analogous to latent heat in solid-liquid-gas phase transitions)


•Friction:

σn

τ

μ = τ/σn

τ

General definition: localized deformation associated with sliding on existing surface (no large-scale extension of surface area, but microscopic contact area decreases).

In pure frictional sliding across fault, the released strain energy is converted to heat and seismic radiation.


•Plastic flow:

General definition: distributed deformation of solid associated with internal motion of defects (“dislocations”) in lattice.

Surface area is conserved. The released strain energy is converted to heat (and some radiation).

Highly enhanced by increasing T, which improves the dislocations mobility, and P.


•Viscous flow:

General definition: distributed deformation associated with fluid-like flow satisfying

E.g., for linear Newtonian viscosity

which in 1D is

Strain energy is converted to heat.

Creep is a form of distributed or localized viscous flow associated with dislocations and diffusion of material

)(ετ &f=

klijklij D ετ &=

εητ &=


•Cataclastic flow:

Mega-breccia Micro-breccia

General definition: distributed brittle deformation associated with motion of rock particles on a large collection of cracks and/or frictional surfaces

Strain and gravitational energy converted to heat (and in dynamic events also radiation and fracturing)


•Viscoelasticity: distributed deformation that is elastic on short time scales and viscous on long ones.

(a) Maxwell viscoelasticity: ησσ

με += &&

1

E v

(b) Kelvin-Voigt viscoelasticity: εημεσ &+=

σσ0

t

εε0

t

Since there is no clear criterion on what are “small” and “large”time scales, most materials belong to this category.

In both Maxwell and KV viscoelasticity, the characteristic time scale for relaxation isT = η/μ.

The response of material at different time scales may be characterized by “Deborah number”D=T/t0, where T is relaxation time and t0 is time scale of interest

E.g., D >> 1 ~ elastic“Standard viscoelasticity” is (a) in parallel with a spring


Brittle failure Ductile flow

localized deformation with fracture, friction and increasing surface area

Sharp tip and/or surface

Stable or unstable (usually unstable)

Failure (stress drop) under small strain

Strong dilatancy: ΔV/V > 0 (“swelling”of the deforming material)

Strong dependency on σn (because of dilatancy and friction)

Weak dependency on T (in the brittle range) jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

Favored by low T, P, e.g., z < 7.5 km

distributed deformation with viscous, plastic, creep (~constant surface area)

No sharp tip and/or surface

Stable or unstable (usually stable)

Ability to sustain large strain without failure

no dilatancy kjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

weak dependency on σnkjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj

strong dependency on T (because of increasing mobility with increasing T)

Favored by high T, P e.g. 15 < z km

•Deformation in the range 7.5 < z < 15 km is mixed plastic-brittle or semi-brittle•The mode of deformation depends strongly on space-time scales!!!

http://www.cacr.caltech.edu/~slombey/asci/solids/fracture/


Brittle - Ductile transition:

frictionalgzWns )(7.0 ρρμστ −≈≈

creep nRTQeA τε /0

−≈&

Scholz2002

This only demonstrates plausibility! Other mechanisms can produce similar distributions. Be aware of non-uniqueness!


Seismicity on a model fault governed by friction and creep (Ben-Zion, 1996)

Can get similar distributions also with rate-state and damage rheology!

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