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Tectonics
Lecture 11
State of Stress in the Crust
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Global distribution of tectonic deformation
ß
Plate boundaries and zones of distributed deformation (after
Gordon, 1994)
Many plate boundaries are so indistinct that they occupy 15% of
the Earth’s surface
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Deformation of tectonic plates
• Strain rates inferred from
summation of Quaternary
fault slip rates (white axes),
and spatial averages of
predicted strain rates (black
axes) given by fitted
velocities
• Fitted strain rate field is a
self-consistent estimate in
which both strain rates and
GPS velocities are matched
by model strain rates and
velocity fields.
GNH7/GG09/GEOL4002 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD
Deformation of tectonic plates
Global strain
rate model
For the global model ~1600 geodetic velocities are currently
used. These velocities comprise mainly of GPS, but velocities
from the SLR, VLBI and DORIS techniques are also used.
Seismic moment tensors from the Harvard CMT catalogue are
taken to infer a seismic strain rate field.
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Diangxiong Fault, Tibet
UCL-Birkbeck
China joint
project:
InSAR edgereflector and
GPS network
around the
Dangxiong Fault
and Quaternary
geology slip rates
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The Tectonic Cycle
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Forces on Plates
•Large meteoritic impacts
May explain sudden changes in rates or direction of plates (e.g.
Scotia arc)
• Slab pull at ocean trenches
Argument against: Once a plate has reached terminal velocity
slab pull is balanced by viscous and frictional forces
• Drag at the base of the plate through mantle convection
Implausible because plates not coupled to mantle. Strain rates at
plate boundaries are up to 109 than in plate interiors.
•Ridge push at mid-ocean ridges from upwelling magma
10x less effective than slab pull but also have gravity slide
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Ridge push / gravity slide
Necessary Zagros
fold force 0.145 kbar
Gravity slide 0.28 kbar
Ridge push 0.3 kbar
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Crustal stress map
Measurements of stress are
usually derived from
displacement. However
there are some more direct
methods such as
measuring stress of
borehole breakouts, and
also methods derived from
seismology, which we will
discuss later.
The stress maps display the orientations
of the maximum horizontal stress SH
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State of stress in the crust
ß At the surface
atmospheric pressure neglected – but can be significant on Venus
free surface
Earth’s surface
air/water – can’t support shear stress
density ρ
rock – only shear stresses in this plane
p = σZ
ß Near the surface
1 vertical + 2 horizontal
principal stresses
lithostatic stress:
p =σz = ρ g z
ß Deeper in the crust
the overburden pressure or
lithostatic stress becomes
increasingly significant
depth z
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Lithostatic stress
p = σZ
ß Lithostatic stress or pressure:
ß overburden pressure
lithostatic stress:
p = σz = ρ g z
Typical density ρ = 2.5 x 103kg.m-3 in the
upper crust
Acceleration due to gravity g = 9.8 m/s2
depth z
The geobaric gradient is 25 MPa/km in
the upper crustal. Density is pressure
and temperature sensitive and so the
geobaric gradient varies according to
tectonic environment.
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Tectonic stress
ß
ß
Deviatoric stress σ’
a Is the amount the total stress deviates from the
mean stress or hydrostatic pressure
a it is a tensor
Differential stress σd
a Is the difference between the max & min
stresses: σd = σmax - σmin
a Often p = σ3 is called the pressure
•
Deviatoric stress tensor = total stress tensor – hydrostatic pressure
•
Deviatoric stress drives deformation of the crust
•
Differential stress = max stress – pressure (-σ3)
•
Differential stress also drives deformation of the crust
•
Both can be considered to be the tectonic stress, σtect
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Seismic / aseismic transition
Strength profile
Shear resistance
Overburden
Brittle
40km 1 kbar
100MPa
Maximum shear strength, maximum
stress drop → big earthquakes
Ductile
Depth
Thermally activated
creep: exp(-H/kT)
Higher strain rate
Low geothermal gradient
Pore fluid pressure
Earthquake locations show the seismic zone is close to the upper
surface of the down-going plate
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Measuring stresses in the crust
Stress
Comment
Pressures
Atmospheric pressure
Sea level
0.1 MPa
Upper crust
0-25 km
0-600 MPa
Core
5,100 km
350 GPa
Ridge push
Red Sea
20-30 MPa
Geodynamics
Ridge gravity slide
Red Sea
30 MPa
Geodynamics
Continental collision
Zagros
15 MPa
Geodynamics
Tectonic stresses
Active regions
Various
10-100 MPa Strain 0.2-0.6 x10-6
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Water in the crust
p = σZ
ß Hydrostat
lithostatic stress:
p= σz = ρ g z
pf = ρw g z
pf – pore fluid pressure
ρw – density of water
The average bulk density of water is
approximately 1.0 x 103kg.m-3 . This
will vary depending on salinity,
temperature and pressure
hydrostat:
depth z
pf = ρw g z
The hydrostat or pore fluid pressure gradient is 10 MPa/km
in the crust. This is about 40% of the lithostatic pressure
The ratio of the pore fluid pressure to the lithostatic pressure is the pore fluid
factor: λv = pf / p
The effective overburden pressure may then be written
peff = p - pf = ρ g z (1 - λv)
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Water in the crust
ß Lithostatic stress vs
hydrostat
The hydrostat or pore fluid pressure gradient
is 10 MPa/km in the crust. This is about 40%
of the lithostatic pressure
Typically the pore fluid factor:
λv = pf / p = 0.4
This is just the weight of the water
column to the rock column.
KTB borehole
Suprahydrostatic gradients are known to occur in tectonically controlled
areas created by fault sealing or by impervious rock layers.
Fluid pressures are commonly greater than hydrostatic during crustal
deformation, particularly in compressional tectonic regions. For example,
east of the San Andreas, fault fluid levels deviate from an initial
hydrostatic gradient to λv values of 0.9 over the depth range of 2-5km .
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remote principal stress
2D stress field
remote principal stress
σ1
fault plane
normal stress
σN
σ2
σS
shear stress
σ1
P
θ
σ2
Resolution of forces and
areas both parallel and
perpendicular to the fault
leads to the following
equations for normal and
shear stress on the fault
plate:
Normal stress σN and shear stress σS
σ N = 1 2 (σ 1 + σ 2 ) − 1 2 (σ 1 − σ 2 )cos 2ϑ
σ S = 1 2 (σ 1 − σ 2 )sin 2θ
Note that: ½ (σ1 + σ2) = σm = mean stress
Local stresses on fault: σ1 > σ2 > σ3 compression positive
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Construction of Mohr stress circle:
shear stress vs. normal stress
σS axis
Maximum shear stress = ½ (σ1 - σ2) when θ = 45o
σS max
P
σS
(σ1 - σ2)/2
2θ
σ2
σN
σm
σ1
σN axis
Any point on circle has coordinates (σN, σS)
where:
(σ1 + σ2)/2
σ N = 1 2 (σ 1 + σ 2 ) − 1 2 (σ 1 − σ 2 )cos 2ϑ
σ S = 1 2 (σ 1 − σ 2 )sin 2θ
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