Stress, Strain, Elasticity and
Faulting
Lecture 11/23/2009
GE694 Earth Systems Seminar
Linear Elasticity: StressStrain Relations
For a linear elastic material, the constitutive relation linearly relates stress and
strain. The constants of proportionality are called “elastic constants”. There are
different elastic constants depending on the form of the stress-strain (i.e.,
constitutive) relation.
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• Example stress-strain measurements:
Linear elasticity below
this load level. Nonlinear
elastic behavior above
this load level (fracture
can occur).
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Axial strain is in the y direction.
Lateral strain is in the x and z
directions.
Stresses in Different Coordinate
Systems and Principal Stresses
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The principal stresses are
a convenient description of
the stress field. There are
maximum, intermediate
and minimum principal
stresses.
These formulas show how to relate the
principal stresses to the shear and
normal stresses. From earthquake
focal mechanisms, the maximum,
intermediate and minimum principal
stresses are called the P, B and T axes,
respectively.
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These three figures show the maximum
and minimum principal stresses for
thrust (top left, cross-sectional view),
normal (top right, cross-sectional view),
and strike-slip (bottom right, map view)
faults. The inward pointing arrows
show the maximum compressive stress
direction (P axis), and the outward
pointing arrows show the minimum
compressive stress direction (T axis).
In all figures, the intermediate
compressive stress direction (B axis) is
perpendicular to the plane of the figure.
This map shows the direction of the maximum principal stress. The symbols show
normal faults (NF), strike-slip faults (SS), thrust faults (TF), or undetermined faults
(U).
This map shows the direction of the maximum principal stress. The symbols show
normal faults (NF), strike-slip faults (SS), thrust faults (TF), or undetermined faults
(U).
This map shows the direction of the maximum principal stress. The symbols show
normal faults (NF), strike-slip faults (SS), thrust faults (TF), or undetermined faults
(U).
This map shows the direction of the maximum principal stress. The symbols show
normal faults (NF), strike-slip faults (SS), thrust faults (TF), or undetermined faults
(U).
This map shows the direction of the maximum principal stress. The symbols show
normal faults (NF), strike-slip faults (SS), thrust faults (TF), or undetermined faults
(U).
Fault Friction and Fault
Movement
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Faults are assumed to be locked by static friction. When the ratio of the
shear stress to the normal stress on a fault overcomes static friction, the fault
slips in an earthquake.
Elastic Rebound Theory
Figure 8-4 shows
Reid’s elastic
rebound theory.
Static friction
holds the fault
until failure is
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Anderson Theory of Frictional
Faulting
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Anderson’s theory shows how to calculate the
normal and shear stress across a fault. If the
ratio of the shear stress to the normal stress
exceeds static friction, the fault moves.
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Data used to estimate
the coefficient of static
friction for rocks.
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Laboratory
measurements show
that rocks fail more
easily under tension
than they do under
compression. Thus,
normal faults form
more easily in the
Earth than thrust
faults. Because of
viscous creep in the
mantle, the rocks tend
to flow rather than
deform elastically and
slip in brittle failure.
The slider-block model of section 8-7 in the textbook is an analog that approximately
describes how faults experience periodic slips due to large earthquakes.
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The solutions given in equations (868) and (8-69). These solutions
describe what is called “stick-slip”
sliding.
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Earthquake Scaling Relations
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Average fault slip increases with fault
rupture length, and therefore
earthquake magnitude and seismic
moment.
Earthquake magnitude and seismic
moment increase with fault rupture
length.
The plot at left shows some average
earthquake scaling relationships.
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Earthquake interaction: The Coulomb Failure Function
A function that measures the enhancement of the failure on a
given plane due to a stress perturbation is the Coulomb Failure
Function (CFF):
CFF   S   N ,
where:
S is the shear stress (- positive in the direction of slip)
N is the normal stress (- positive in compression)
M is the coefficient of friction

Failure on the plane in question is enhanced if CFF is
positive, and is delayed if it is negative.
Earthquake interaction: The Coulomb Failure Function
The figures above show the change in the fault-parallel shear
stress and fault-perpendicular normal stress, due to right-lateral
slip along a dislocation embedded in an infinite elastic medium
Earthquake interaction: The Coulomb Failure Function
Earthquake interaction: Stress shadows
The 1906 Great California stress shadow:
Stein, 2002
So the CFF concept works not only for positive, but also for
negative stress change.
Seismicity and Faults
1992 Landers and 1999 Hector
Mine, California Earthquakes
Fault ruptures (solid lines) and
maximum stress directions (lines
with circles) for the right-lateral
strike-slip Landers and Hector
Mine faults.
Slip (top rows), stress drop
(middle rows) and static
friction values (bottom rows)
for (a) the Lander
earthquake and (b) the
Hector Mine earthquake.
Earthquake interaction: Multiple stress transfers - The Landers
and Hector Mine example
Maps of static stress changes
suggest that the Landers
earthquake did not increase the
static stress at the site of the Hector
Mine rupture, and that Hector Mine
ruptured within a “stress shadow”.
Kilb, 2003
Earthquake interaction: Multiple stress transfers - The Landers
and Hector Mine example
This map shows the
change in CFF caused
by the Landers quake on
optimally oriented planes
at 6km depth. The arrows
point to the northern and
southern ends of the
mapped surface rupture.
Figure downloaded from
www.seismo.unr.edu/htdocs/WGB/Recent.old/HectorMine
Earthquake interaction: Multiple stress transfers - The Landers
and Hector Mine example
• Most Landers aftershocks in the
rupture region of the Hector Mine
were not directly triggered by the
Landers quake, but are secondary
aftershocks triggered by the M 5.4
Pisgah aftershock.
• The Hector Mine quake is,
therefore, likely to be an aftershock
of the Pisgah aftershock and its
aftershocks.
Felzer et al., 2002
Earthquake interaction: Aftershock triggering
Maps of CFF calculated following major earthquakes show a
strong tendency for aftershocks to occur in regions of positive
CFF.
The Landers earthquake (CA):
King and Cocco (2000);
Stein et al., 1992.
Earthquake interaction: Aftershock triggering
The Homestead earthquake (CA):
King and Cocco (2000).
Earthquake interaction: The domino effect
Example from California:
Figure from www.earthquakecountry.info
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Earthquake interaction: The domino effect
Example from the North Anatolia Fault (NAF):
Figure from Stein et al., 1997
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Northeastern seismicity, October 1975 to September 2009.
Earthquake interaction: Remote aftershock triggering
NÝIzmit + 10 days   NÝIzmit - 100 days 
NÝ1985 - 2002
The Mw7.4 Izmit (Turkey):

Mw5.8
Two weeks later
Earthquake interaction: Remote aftershock triggering
The decay of M7.4 Izmit
aftershocks throughout Greece
is very similar to the decay of
M5.8 Athens aftershocks in
Athens area (just multiply the
vertical axis by 2).
Earthquake interaction: Dynamic triggering
CFF(t)   S (t)   N (t) ,
• The magnitude of static
stress changes decay as
disatnce-3.
• The magnitude of the peak
dynamic stress changes
decay as distance-1.
• At great distances from the
rupture, the peak dynamic
stresses are much larger
than the static stresss.
Figure from Kilb et al., 2000
Earthquake interaction: Dynamic triggering
Instantaneous triggering
Time
No triggering
Time
Earthquake interaction: Dynamic triggering
Indeed, distant aftershocks are observed during the passage of
the seismic waves emitted from the mainshock rupture.
Izmit aftershocks in Greece.
Brodsky et al., 2000
Earthquake interaction: Dynamic triggering
• Dynamic stress changes trigger aftershocks that rupture during
the passage of the seismic waves.
• But the vast majority aftershocks occur during the days, weeks
and months after the mainshock.
• Dynamic stress changes cannot trigger “delayed aftershocks”,
i.e. those aftreshocks that rupture long after the passage of the
seismic waves emitted by the mainshock.
• It is, therefore, unclear what gives rise to delayed aftershocks in
regions that are located very far from the mainshock.
Further reading:
• Scholz, C. H., The mechanics of earthquakes and faulting, NewYork: Cambridge Univ. Press., 439 p., 1990.
• Harris, R. A., Introduction to special section: Stress triggers,
stress shadows, and implications for seismic hazard, J. Geophys.
Res., 103, 24,347-24,358, 1998.
• Freed, A. M., Earthquake triggering by static, dynamic and
postseismic stress transfer, Annu. Rev. Earth Planet. Sci., 33, 335367, 2005.