Non-Linear Radiation Damping: A New Method for Dissipating Energy in Dynamic Earthquake Rupture Simulations


 Dynamic earthquake rupture simulations are used to understand earthquake mechanics and the ground shaking that earthquakes produce. These simulations can help diagnose past earthquake behavior and are also used to generate scenarios of possible future earthquakes. Traditional dynamic rupture models generally assume elastic rock response, but this can lead to peak on-fault slip rates and ground shaking that are higher than those inferred from seismological observations. Some have approached this challenge using inelastic off-fault rock behavior to dissipate energy, but the addition of inelasticity can make it difficult to select parameters and establish suitable initial conditions, and increases the model’s complexity and computational cost. We propose a new method that works by adding a nonlinear radiation damping term to the friction law, with the surrounding rocks remaining linear elastic. Our new method results in lower peak slip rates, reduced seismic radiation, and an increasing slip-weakening critical distance with increasing rupture propagation distance, all within a linear elastic model. In addition, it is easy to implement.

Our Hayward dynamic rupture simulations used linear elastic material properties.
It's easy to set up initial conditions in elastic models: • Can assign arbitrary shear and normal tractions on the fault.
• No need to know absolute stress tensor in the model volume.
• No need for gravity or fluid pressure.
But: Elastic models can produce unrealistically high slip rates and ground motions.
Traditional method for setting initial stress in viscoplastic models: • Must specify absolute stress tensor throughout the model.
➢ Stress tensor appears in viscoplastic constitutive law.
• Fault tractions are determined by stress tensor, and must be compatible with friction parameters.
• Initial stress tensor must be compatible with viscoplastic parameters.
• Must include gravity and fluid pressure.
• Initial stress must be in static equilibrium.

Difficulties of Adding Viscoplasticity
Adding viscoplasticity to something like our Hayward model poses difficulties: • 3D heterogeneous velocity and density structure, with gravity → It's hard to find an initial stress tensor in static equilibrium.
• 3D fault geometry → It's hard to find an initial stress tensor that produces acceptable tractions on the fault.
• Stress in the Earth's crust and viscoplastic parameters are poorly known.
• End result: A model with lots of free parameters and initial conditions, that are poorly constrained, and yet difficult to specify.
• Can increase the computational cost by as much as a factor of 3.
Our goal: Find a way to add the effects of viscoplasticity to our model, that avoids these difficulties, and retains the simplicity and efficiency of a linear elastic model.

Joe Andrews' Approach (JGR 2005) -"Velocity Toughening"
In a linear elastic model, impose a maximum slip rate:

≤ max
Joe's implementation is to modify the friction law so that, when = max , the friction law becomes (in 2D) where elastic is the shear stress induced by the elastic stress tensor (not including inertial forces). This reduces the acceleration to zero, leaving constant. (In 3D there is an additional complication due to the possibility of rake rotation, but the concept is the same.) Joe showed that velocity toughening could, in some ways, make a linear elastic model behave as if it had off-fault yielding. But there are several problems with this approach: 1. It produces very strange-looking slip histories, where the slip rate is constant from some period of time.
2. It is difficult to give a physical interpretation.
3. The separation of elastic and inertial forces is not how friction usually works.
(Friction usually responds to the sum of elastic and inertial forces.)

Radiation Damping
Radiation damping is a standard technique in quasi-static earthquake simulators. It compensates for the lack of dynamics, by adding a term to the friction law: damping = 2 = shear modulus = shear wave velocity = slip rate Notice this is linear in .
Physically, when fault slip occurs, the inertia of the surrounding rock produces a reaction force that opposes further slip. Quasi-static earthquake simulators don't have inertia, and the absence of that reaction force produces slip rates that are too high.
The radiation damping term supplies the reaction force, which is otherwise not present in a quasi-static model. Including it allows a quasi-static model to behave, in some respects, as if the model contained dynamics.

Non-Linear Radiation Damping
In a viscoplastic model, off-fault yielding reduces the magnitude of the stress tensor, thereby reducing the shear traction acting on the fault. The effect is the same as if the off-fault yielding produced an additional reaction force that opposes further slip, above and beyond the reaction force of inertia.
Our idea is to take a linear elastic dynamic rupture model and add a non-linear radiation damping term to supply that additional reaction force. This allows a linear elastic model to behave, in some respects, as if the model contained viscoplastic yielding. • Linear slip-weakening friction.
• Gravity and fluid pressure.
The setup is the same as the SCEC/USGS benchmarks, except that: • We create "soft" boundaries at the lateral ends of the fault, so the rupture stops spontaneously before it reaches the ends of the fault. This is done by increasing the length of the fault from 40 to 50 km, and imposing increased frictional cohesion near the ends.
• We reduce the frictional cohesion near the surface so that the rupture can reach the surface in both elastic and viscoplastic cases.

Sample Runs with Non-Linear Radiation Damping -Based on TPV26v2 and TPV27v2
Plots show slip rate at distances of 5 km (black), 10 km (red), 15 km (green) and 20 km (blue) from the hypocenter. • The x-axis is slip in m.
• The y-axis is shear stress in MPa.
• The apparent critical slip distance is where the shear stress reaches its final value.
• The apparent fracture energy is the area under the curve and above the final value.
• The slip-weakening critical distance in the friction law is fixed at = 0. • Plots show three components of particle velocity at two locations on the Earth's surface near the fault (triangles in figure above).
• The elastic case has highest PGV (peak ground velocity).
• The viscoplastic case has lower PGV.
• The two radiation damping cases have even lower PGV.