The role of frictional plasticity in the evolution of normal fault systems

The problem

Where the crust is stretched, it breaks along normal faults. In each earthquake the hanging wall, the block above the fault, drops relative to the footwall below it. New faults keep forming as a rift grows (Jackson and McKenzie, 1983). Field studies indicate that slip summed across the known faults of a rift does not always account for the stretching rate measured by geodetic surveys (Villamor and Berryman, 2001; Wallace et al., 2004). Other deformation, or faults not yet found, must take up the shortfall.

The western Gulf of Corinth in Greece is stretching at 12 to 14 millimetres (mm) per year (Clarke et al., 1997; Briole et al., 2000; Floyd et al., 2010). There, new faults have formed in the hanging walls of older ones (Jackson, 1999; Goldsworthy and Jackson, 2001). Our goal was to find how frictional plasticity contributes to this evolution. Frictional plasticity is permanent deformation of the crust by sliding on many small fractures. We treat it as a possible precursor to a new fault.

What we did

We built a two-dimensional finite element model of a slice of crust 60 km wide and 25 km deep, with its sides pulled apart at a combined 12 mm per year. The model rock deforms elastically until it reaches its failure limit, and then by frictional plasticity. In the hotter lower crust it creeps slowly.

The crust contains one fault, dipping at 50° to a depth of 15 km. We locked the fault for 220 years and then lowered its friction coefficient to 0.2 for 60 seconds to represent an earthquake. We tallied the split of the applied extension between fault slip, elastic strain, frictional plasticity and creep, the extension budget.

What we found

In the upper 5 km, over a 220-year cycle, the 50° fault takes up about half of the extension and frictional plasticity takes up the rest.

In our model, repeated earthquakes on a low-friction fault keep the crust next to it up to 10 MPa (megapascals, a unit of stress) below its failure limit. We call this region a stress shadow. The shadow is widest near the surface and extends further into the footwall. This is because the footwall moves up and away from the fault in each earthquake, while the model sides move by a negligible amount during the 60-second slip. The shallow footwall is therefore compressed horizontally. This compression lowers the stress there further below the failure limit.

Frictional plasticity is concentrated in bands about 1 km wide that dip at about 45°. Most form in the hanging wall outside the shadow, dipping the same way as the main fault. These bands take up dip-slip in narrow zones. If they lengthen each time they slip, they can be regarded as proto-faults. Whether they become faults is speculative. Their modelled strain rate is equivalent to a slip rate on the order of 0.15 mm per year, about one eightieth of the rate at which the model is stretched.

The fault also rotates to a shallower dip. Between earthquakes, vertical thinning of the stretching crust flattens the fault. During an earthquake, the rotation may come from the footwall at 6 to 9 km depth expanding as its stress drops, pushing on the hanging wall. Together, in almost equal parts, the two periods rotate the fault at about 23.5° per million years. We expect this rate to slow as the fault rotates out of the range in which it can slip.

If both rates stayed constant, after 100,000 years the fault would have rotated from 50° to roughly 48°, and a proto-fault would have an offset on the order of 15 m. These illustrative figures are not in the paper.

We suggest that a young fault takes up only part of the extension, and that proto-faults grow in its hanging wall to take up the rest. Older faults may eventually rotate too far to slip. The shortfall would then grow and new proto-faults would form. Our model is consistent with faulting migrating into the hanging walls of existing faults, as in the western Gulf of Corinth.

If we take the edge of the shadow as the 1 MPa contour, the model suggests a minimum spacing of about 2 km between hanging-wall faults. This is slightly smaller than the 5 to 8 km average in the Gulf (Rohais et al., 2007). However, direct comparison with field studies is not possible, because the size of the shadow depends on the thickness of the brittle crust and on fault dip. The brittle crust there is 10 to 12 km thick (Briole et al., 2000), against 6 to 8 km in our model. We would expect a thicker brittle crust to give larger shadows and wider spacings.

Limitations

The model is a single two-dimensional slice through one fault. The fault is switched between locked and slipping at prescribed times. The model therefore cannot address how earthquakes start. It leaves out the weakening of rock as it deforms (strain softening). As a result the width of the shear bands depends on the size of the model elements. We offer the sequence above as an illustration of processes that may operate in specific places such as the western Gulf of Corinth. Faulting cannot migrate into hanging walls indefinitely unless the zone of extension moves too. Where the zone of extension is stationary, the stress shadow around inactive faults may recover and allow faulting to return to older footwalls.

Glossary

A normal fault is a fault where the crust is being stretched and the upper block slides down the fault plane.

The hanging wall is the block of crust above a dipping fault, and the footwall is the block below it.

A fault dips at the angle it is inclined below horizontal. Dip-slip is movement up or down the slope of a fault.

A geodetic survey repeatedly measures the position of points on the ground.

Frictional plasticity is permanent deformation of rock by sliding on many small fractures.

A proto-fault is a band of concentrated shearing that may develop into a new fault.

A stress shadow is a region near a fault where repeated earthquakes keep the stress below the failure limit.

The extension budget is the split of the applied stretching between fault slip, elastic strain, frictional plasticity and creep.

Elastic strain is stretching or squeezing of rock that is reversed when the stress is removed.

The failure limit is the stress at which rock starts to slide on its fractures.

Creep is slow, continuous flow of hot rock at depth.

Strain softening is the weakening of rock as it deforms.

A finite element model solves the equations of deformation by dividing the crust into many small elements, here about 10,000.

A friction coefficient is the ratio of the force needed to slide a fault to the force pressing it shut.

A megapascal (MPa) is a unit of stress, one million newtons per square metre.