Complex Systems

An earthquake can be viewed as either an instability in frictional contact or the growth of a shear crack.  We model the crust of the earth as an elastic solid that can support an interesting variety of wave motions as well as static stress and deformation fields.  Within this crust are fault planes, zones of weakness along which previous earthquakes have occurred.  Due to the large scale tectonic motion of plates, large stresses accumulate within the ground.  For example, here in Southern California, the Pacific and North American plate slide past each other along the San Andreas fault (you can imagine the fault as a vertical plane of contact extending from the surface down to about 10 or 15 km – see Figure 1).  Below this depth, the increased temperature and pressure allow the rocks to move in a ductile manner that prevents the build-up of large stresses.  But closer to the surface, the rocks are brittle solids that are being compressed by the enormous weight of the overlying rocks (something you might be familiar with if you've been buried in the sand at the beach!).  This compressional stress will prevent the two sides of the earth from opening (which is what happens in a tensional crack).  Since the two sides of the fault are always in contact, there will be a frictional force between them.

Figure 1: A fault, shown in gray, is a roughly planar zone of weakness in the crust.  Tectonic motions of the earth’s plates drive the two sides of the fault past one another, represented here by the arrows.  The yellow surface denotes the earth’s surface.

Now imagine trying to force the two sides to slide past each other.   For the San Andreas, the western Pacific plate (on which Santa Barbara sits) is moving northwest relative to the eastern North American plate (on which most of the country sits) at a rate of about 30-40 mm/yr.  This motion increases the shear stress on the fault over the course of decades or centuries, during which the fault is still locked (see Figure 2). 

Figure 2: Top view of the geometry shown in Figure 1.  The fault is the horizontal line through the center, with the blue arrows representing frictional forces that keep the sides of the fault locked.   The slanted vertical lines indicate the shear displacements created by tectonic loading.

At some critical level, the force on the fault will exceed the static frictional force, and the two sides will start to slide with some dynamic frictional resistance (less than the frictional value).  From a fracture point of view, there is a cohesive force that exists between the two sides of the fault, and the stress must be large enough to overcome this.  The discontinuity in the relative displacement is called slip.  When this process occurs rapidly, it converts some part of the released strain energy into seismic waves (see Figure 3).

Figure 3: Material in the fault zone fails (or static friction is exceeded) and the fault begins to slip.  Physically, we can view this process as the application of shear forces on the fault that negate the static friction, as represented by the red arrows.  This releases elastic waves, indicated by the expanding green circles.

Some of these elastic waves radiate away to generate the ground shaking we experience, while others travel along the fault to the locked region ahead of the crack edge.  These waves carry the shearing force to the locked section and drive it toward failure.  In this way, the crack expands at a velocity comparable to the speed of elastic waves, about 4-6 km/s!   That’s a good deal faster than the rate at which the plates move.  As you can see, this process is cyclical, and can lead to a rapidly expanding rupture (see Figure 4).

Figure 4: The dynamic rupture process.  The green bubbles show the flow of energy within the system.  The yellow and orange boxes show the process by which the energy conversion occurs.

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The strength of faults (i.e., their resistance to slip) varies considerably on almost almost every scale. When a propagating rupture encounters some local variation in strength, its shape and speed are affected and it radiates energy away as seismic waves. The extreme heterogeneity of real faults makes it difficult to decipher the ground shaking. As a start, we study the idealized problem of a single, circular high strength region known as a barrier. Figure 1 and this movie show the rupture process on the fault plane for this problem. This is reported in Dunham et al. (2003).

Figure 1: Consecutive snapshots (top to bottom) of the fault plane, showing a rupture breaking through a high strength barrier. The color scale measures slip velocity (warm colors mean faster sliding), and is renormalized in each snapshot. The rupture propagates to the right and is initially delayed by the barrier (which requires four times more energy to break than the surrounding fault). In contrast to what might be expected, the rupture actually shoots forward ahead of where it would have been in the absence of any perturbation! This is clearly shown in the middle image of the right column.

Why does the rupture temporarily speed up? The answer lies in understanding the effect of rupture front curvature. As described previously , the region behind the rupture front where the stress is released is the source of elastic waves. These waves diffract off of the moving crack edge. If the crack edge is curved, then the waves become focused.This is illustrated for a simple crack edge geometry in Figures 2 and 3 below.

Figure 2: Consider an increment of crack growth when the crack edge is locally concave. Each point behind the crack edge is a source of elastic waves, each of which is represented with a red circle (wave velocity is assumed isotropic for simplicity and only one wave type is shown). Focusing toward the center of the barrier is evident.

Figure 3: Similar to the above figure, but after the waves have converged in the center of the barrier. The center of the barrier is the point of maximum focusing. The pattern of wavefronts matches that observed in the numerical models of the barrier.

Another distinctive feature of the simulation is the set of nearly elliptical slip pulses that radiate from within the barrier. To understand these better, we further idealized the barrier to a point and neglected any diffractions off of the crack edge. This made the problem analytically tractable and we found an exact expression for the transient displacement field anywhere in the surrounding medium or on the fault. This is reported in Dunham (submitted 2004 to J. Mech. Phys. Solids).

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We have recently been focusing on an interesting class of ruptures in which the propagation velocity of the rupture front exceeds the shear wave speed. While the theory for such ruptures has been around since the early 1970s, only in the past few years has solid experimental and observational evidence emerged. Remarkably, the same phenomenon occurs at the cm scale in laboratory fracture experiments (by Rosakis and coworkers at Caltech) as in the earth on the 10 km scale!

We have studied a simple model of a 2D rupture pulse propagating at a constant velocity in order to characterize the type of ground shaking that will occur near faults. Figure 1 shows the velocity field for a subshear rupture, and Figure 2 is the same for a supershear rupture.

Figure 1: Velocity field for a subshear rupture. You are basically seeing how the ground (e.g., the earth's surface) moves as the rupture sweeps by. The color scale measures the amplitude of the ground's velocity and the arrows point the direction of its motion. The zone of active slip is marked by the solid black line and the red line shows the region when the material failure process occurs. Note that the ground shakes predominantly in the direction perpendicular or normal to the fault, consistent with most near-source records of real earthquakes.

Figure 2: Same as Figure 1, but for a supershear rupture. In this case, the shear waves radiate from the slip zone, forming a planar Mach front. A seismogram capturing the passage of the shear waves contains an exact record of the slip velocity history on the fault! Obviously good for seismologists interested in the source process, these ruptures potentially produce large amplitude shaking at distances quite removed from the fault.

Further discussion of the near-source ground motion from supershear ruptures can be found in Dunham and Archuleta (submitted 2004 to GRL)

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In a set of numerical experiments done by Dunham et al. [2003], it was found that in a three-dimensional medium, strong regions (barriers) can produce strong ground motion.  In the figure to the left, rupture velocity is shown on the fault plane at successive steps in time.  The rupture front hits a circular region of higher yield stress, which first stores energy as the rupture front surrounds it.  As energy is focused inward, the barrier fails catastrophically, releasing a strong pulse of energy. To investigate the ground motion produced by this complex dynamic process, we first construct a reduced barrier model with a constant rupture velocity.  Using a 3D finite-difference method, we analyze the effect of barrier radius, strength, and depth, as well as the additional diffraction effects introduced by a time delay before the barrier breaks.

We follow the method of Andrews [1985] to solve the barrier problem given a constant rupture velocity.  Unlike in a kinematic model, this does not constrain the slip-time function of points of the fault.  Rather, it forces slip to be zero ahead of a rupture front moving at constant velocity.  Frictional strength is no longer a function of slip as in a slip-weakening friction law.  Instead, for each point it falls linearly with time until it reaches a constant value below the prestress, as shown in the figure.  This method constrains the rupture velocity to be constant.

We parameterize the barrier problem by R, the radius of the barrier, tau0, the stress drop of the surrounding fault, taub, the additional stress drop in the barrier, and the depth d of the barrier.

For a small barrier, we expect the ground motion of our reduced model to be similar to the superposition of a homogeneous rupture and a point source, with the added effects of diffraction off of the crack edge. After subtracting off the displacements for a homogeneous rupture, this model cleanly shows that all components of additional displacement are proportional to R2 and taub, for all points on the surface, at all times.  The scaling relationship for depth is more complicated, but this parameter has the most influence directly above the hypocenter.  In addition, as expected, this model shows that rise time for surface displacement is not a function of taub. Rise time for surface displacement increases with R and d, most notably in the forward direction from the barrier. The first two columns of the figure on the right show fault-parallel displacements for the free surface for a homogeneous rupture (leftmost column) and the barrier rupture (middle column) at successive times.  The bottom edge is the fault plane in these figures.  The right-most column shows velocity on the fault plane for the barrier model.

To make our model more realistic we next delay the breaking of the barrier by a time  tb.  Numerical work by Dunham shows that this delay time is given by tb=(2R/cs)(1+aG/G0), where R is the radius of the barrier, cs is the shear wave speed, G is the fracture energy, and G0 is the energy release rate.  The constant a=0.6 was fit numerically using a slip-weakening friction law, as shown in the figure to the left.

This delay time makes the ground motion more dramatic: as the rupture front passes through the unbroken barrier, the stress increases, particularly at the edges of the unbroken region (see figure to right).  When the barrier finally breaks, it breaks more violently than in the first model without the time delay.

Above: The stress increase in the locked barrier as the rupture front passes.

The barrier time delay also changes the diffraction effects from the first model -- for the crack edge is further ahead.  The figure to the left shows the different wave fronts we can expect. The black circle shows the origin of a point source.  The lines at the left side of the diagram show the velocity waves caused by the source: green curves are p-waves, blue are s-waves, and red are Rayleigh waves.  To the right of the crack edge, the fault surface is unbroken, so that the curves shown are stress waves.

The middle column of the figure to the right shows the fault-parallel velocity at various times for our second model, with the barrier delay time.  The last column shows velocity on the fault for the second model.   Unlike in our original model, the barrier, initially locked as the rupture front passes, arrests the ground motion, before a larger pulse from the breaking of the barrier arrives.

These models are not fully dynamic, as we are constraining the rupture velocity to be constant.  The next step is to examine the fully dynamic problem, which may include supershear transients as seen by Dunham et al.  The 1984 Morgan Hill earthquake may be a good example of an earthquake that can largely be characterized in terms of these barrier models. The kinematic inversion of Beroza and Spudich [1988] shows that most of the slip in the Morgan Hill earthquake was concentrated in a small portion of the fault. The slip in this region is believed to be accountable for large, late pulses in several of the seismograms.  Furthermore, their inversion shows a rupture delay in this region -- further evidence of a barrier.  Finally, large fault parallel motions at the Coyote Lake Dam station may be evidence of supershear motion, as seen in the work of Dunham et al. [2003].

Above: The slip and rupture time on the fault for the 1984 Morgan Hill earthquake, as calculated in the kinematic inversion of Beroza and Spudich.

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References:

Andrews, D. J., Dynamic Plane-Strain Shear Rupture with a Slip-Weakening Friction Law Calculated by a Boundary Integral Method, BSSA, 75, 1-21, 1985.

Beroza, G. C., and P. Spudich, Linearized Inversion for Fault Rupture, J. Geophys. Res., 93, 6275-6296, 1988.

Dunham, E. M., P. Favreau, and J. M. Carlson, A Supershear Transition Mechanism for Cracks, Science, 299, 1557-1559, 2003.