Stress, Strain, Elasticity and Faulting Lecture 11/23/2009 GE694 Earth Systems Seminar.

Stress, Strain, Elasticity and Faulting

Lecture 11/23/2009

GE694 Earth Systems Seminar

Linear Elasticity: Stress-Strain 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:





Axial strain is in the y direction. Lateral strain is in the x and z directions.

Linear elasticity below this load level. Nonlinear elastic behavior above this load level (fracture can occur).

Stresses in Different Coordinate Systems and Principal Stresses

The principal stresses are a convenient description of the stress field. There are maximum, intermediate and minimum principal stresses.





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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.










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).





Fault Friction and Fault Movement

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









Anderson Theory of Frictional Faulting

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.











Data used to estimate the coefficient of static friction for rocks.

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.












The solutions given in equations (8-68) and (8-69). These solutions describe what is called “stick-slip” sliding.



Earthquake Scaling Relations





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.



A function that measures the enhancement of the failure on a given plane due to a stress perturbation is the Coulomb Failure Function (CFF):

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 ispositive, and is delayed if it is negative.

Earthquake interaction: The Coulomb Failure Function

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CFF = Δσ S − μΔσ N ,


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


The 1906 Great California stress shadow:

Stein, 2002

So the CFF concept works not only for positive, but also for negative stress change.

Earthquake interaction: Stress shadows

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

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



• 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).

Example from California:

Figure from www.earthquakecountry.info

Earthquake interaction: The domino effect





Example from the North Anatolia Fault (NAF):

Earthquake interaction: The domino effect

Figure from Stein et al., 1997

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Northeastern seismicity, October 1975 to September 2009.

Earthquake interaction: Remote aftershock triggering

The Mw7.4 Izmit (Turkey):

Mw5.8Two weeks later

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˙ N Izmit + 10 days( ) − ˙ N Izmit - 100 days( )˙ N 1985 - 2002( )

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

Figure from Kilb et al., 2000

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CFF(t) = Δσ S (t) − μΔσ 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.


Str

ess

Time Time

Instantaneous triggering No triggering


Brodsky et al., 2000

Indeed, distant aftershocks are observed during the passage of the seismic waves emitted from the mainshock rupture.

Izmit aftershocks in Greece.


• 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, New-York: 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, 335-367, 2005.

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Stress, Strain, Elasticity and Faulting Lecture 11/23/2009 GE694 Earth Systems Seminar.

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