GEO 5/6690 Geodynamics 15 Oct 2014

© A.R. Lowry 2014Read for Wed 22 Oct: T&S 105-130

Last Time: RHEOLOGY

Dislocation creep is sensitive to: • Temperature (as exp(C/T)) • Lithology (pyroxene & olivine strongest; feldspar intermediate; quartz weakest) • Water (and possibly other volatiles, e.g., CO2?)

Diffusion creep is sensitive to each of those plus • Grain size

A Yield Strength Envelope represents the deviatoric stress that can be supported by the lithosphere as a function of depth, and assumes a brittle field rheology (including friction coefficient, pore fluid pressure) plus strain rate

Temperature variations in the U.S. (from P-velocity) indicate temperature is not the only control on lithosphere stability!

Next Journal Article Reading:For Monday Oct 20: Pérez-Gussinyé et al. (2009) Effective elastic thickness of Africa and its relationship to other proxies for lithospheric structure and surface tectonics. Earth Planet. Sci. Lett. 287(1-2) 152-167.

Brittle-field failure & the “Seismogenic Zone”

So what about that top part of the Yield Strength Envelope?

For large-scale deformation in the real Earth, other important rheologies include brittle (frictional) failure in rock fracture & fault slip, described by Byerlee’s Law:

Place a rock sample with an existing fracture in a lab apparatus, apply a stress normal to the fracture plane N, then determine how much shear stress is required to get it to slip; frictional strength is defined by a friction coefficient :

€

=N

Byerlee’s Law

For dry rock, no gouge, low temperature… this is insensitive to rock type!

Lab experiments & borehole stress tests suggest ~ 0.6…But dynamical models indicate long-term nearer 0.1-0.2!

Using Byerlee’s law (and assuming ubiquity of fractures),can define a depth-dependent failure law with frictionalyield strength as (e.g., Sibson, Nature 1974):

where = P/gz and P is pore fluid pressure, is mass density of rock, g is gravitational acceleration, z is depth

This is where the asymmetry in thebrittle-field part of the YSEarises!

€

= 2 +1 − μ( )−2

−1 ⎡ ⎣ ⎢

⎤ ⎦ ⎥ρgz 1− λ( )

€

= 1− μ 2 +1 − μ( )2 ⎡

⎣ ⎢ ⎤ ⎦ ⎥ρgz 1− λ( )

in compression ( < 0);

in extension ( > 0)

0

z

Of course the bottom part of the Yield Strength Envelopereflects a creep flow law, so it requires assumptions aboutlithology, geothermal gradient, water fugacity & grain size:

Note however this also requires some assumption aboutthe strain rate !

Typically we assume a constant strain rate of ~10-14–10-16 s-1.But in reality, strain rate is determined by the stress forcingvia the constitutive law (so if we know, everywhere in 3D,the density structure and therheology, we could calculate ormodel based on that). The YSE was first describedand used by Goetze &Evans (GJRAS 1979).

€

˙ ε

0

z

€

= ˙ ε A−1d m fH2O−r

( )1

n expQ + PV

nRT

⎛

⎝ ⎜

⎞

⎠ ⎟

€

˙ ε

0

z

To first order, the distribution of earthquakes with depthdepends on the YSE, andthe “seismogenic depth”will correspond to the brittle-ductile transition.

brittle =seismogenic

ductile

z

brittle =seismogenic

ductile

BUT earthquake focaldepths will map to thebrittle-ductile transitionONLY if ambient stresseverywhere exceeds the yield strength!!!

max EQfocal depth

But important to note: Frictional slip is dynamical and also depends on rate, state

FavorsStableSliding

FavorsStick-Slip

Fault friction and slip velocity:

• Two frictional constants a and b plus a length scale Dc

• a – b < 0: friction decreases with increasing slip velocity ( unstable)

• a – b > 0: friction increases with increasing slip velocity ( stable)

• a, b, Dc depend on temperature, rock type, pore fluids, gouge properties

• State variable (depends on history of slip, evolution of contact surface)

• Normal and shear stress on the fault

Depend on:

Hence friction & slip vary nonlinearly in both space and time!

€

dΘ

dt= −

V

Dc

Θ + b lnV

V *

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

€

t( ) = μ 0 + a lnV

V *

⎛

⎝ ⎜

⎞

⎠ ⎟+ b ln

V *Θ

Dc

⎛

⎝ ⎜

⎞

⎠ ⎟

Cumulative Slip (m)

Pre-Slip

Earthquake

After-Slip

J. GEOPHYS. RES., 105(E10), 2000

b – a = 0

b – a = 0

b – a < 0

b – a > 0

b – a > 0

A San Andreas simulation with rate-state constitutive relations…

Predicts geodetic behaviors that we observe.(But what form does this part take?)

0

z

brittle =seismogenic

ductile

≠

Result: Earthquakes are distributed only within velocity-weakening zone!