7 .3 Discussion
8.1 Earthquake Source
We need to make only a few, simple modifications to the model of the earthquake source to add the ability to simulate the earthquakes using dynamic failure. Instead of specifying the displacements at the slip degrees of freedom, we use the friction model to specify the forces actmg on the slip degrees of freedom. We must also include the stresses from the surrounding region that act on the fault, '\Yhich we call the tectonic otrcssca because 11 significnnt portion comes from plate tectonics.
We assume that the coefficient of friction is a function of slip distance and slip rate, and possibly, a number of state variables. We use the usual definition of slip rate, i.e., the magnitude of slip velocity. When sliding occurs on a plane, the definition of slip distance depends on the length scale of the surface asperities (surface roughness) that create the friction. If the asperities are large compared to the distance over which slip occurs, then an appropriate definition of slip distance is the magnitude of the distance a point slides from its original position. This definition allows the slip distance to remain constant if sliding occurs along the circumference of any circle centered at the point where sliding begins. On the other hand, if the asperities are small compared to the distance over which sliding occurs, then the slip distance should increase independently of the slip path, and an appropriate definition of slip distance is the total distance over which sliding has occurred. We will assume that the asperities are small compared to the slip distance and use the total sliding distance as the slip distance. Regardless of how we choose to define the slip distance, the friction force always acts in the opposite direction of the sliding.
8.1.1 Governing Equations with Friction
We replace the force vector in the governing equation, equation (2.2), with the difference between the friction force vector, {Ff}, and the vector of tectonic forces, {Ft}, as shown in equation (8.1).
[M]{u(t)}
+
[C]{u(t)}+
[K]{u(t)} = {Ft(t)}-{F1(D(t),D(t))} (8.1)The minus sign in front of the friction force vector indicates that we choose to explicitly include the dissipative nature of the friction force in the governing equation, so that the vector { F1 } acts in the direction of sliding. We will discuss the formulation of the vector of tectonic forces in section 8.1.2.
Following the same procedure that we used for the prescribed ruptures discussed in section 2.2, we integrate the differential equation using the central-difference scheme. The expression for the displacement at time t
+
At is( 1 1 ) .
At2 [M]
+
2At[C] {u(t
+
~t)} = {Ft(t)} {F1(D(t),D(t))}+ (;t2 [M] - [KJ) {u(t)} (8.2)
- ( -1
[M] - -1
[CJ) {u(t - At)}.
At2 2At
We compute the friction at time t assuming that we know the slip rate at time t. In the central-difference scheme the velocity at time t depends on the displacement at time t
+
At, so that computing the slip rate at time t requires knowing the slip at time t +At, which we do not know.To remedy this difficulty, we assume that the time step is small enough so that the slip rate does not change significantly in a single time step. This approximation may cause problems if the slip rate exhibits a strong influence on the coefficient of friction. Fortunately, we do not use friction models with this feature. Thus, we use the slip rate at time t - At, instead of the slip rate at time t, to compute the friction force at time t. Equation (8.3) gives the amended version of the expression for the displacement at time t
+
At.(;t2[M] +
2~t[cJ)
{u(t+At)} {Ft(t)} {F1(D(t),D(t At))}+ ( /1~ 2
[M] [K]) { u(t)} (8.3)- ( -1
[M] - -1
-[c]) {u(t-At)}
At2 2At
8.1.2 Forces on Slip Degrees of Freedom
We must transform the initial tractions applied on the fault surface into forces acting on the slip degrees of freedom. We specify the initia,l tractions on the foult surface u:sing the :spatial interpolation procedure described in section 2.5. At each node on the fault, we interpolate from the given initial tractions and convert the tractions to forces using the node's tributary a.ma on thP. fa11lt pla.nP.. WA assume that the fault is in equilibrium and apply the forces equally to both sides of the fault. We transform the forces into the slip coordinate frame using the transformation matrix [Tslip] given by equation (2.19). Equation (8.4) shows the simplified expression for the force vector applied at the degrees of freedom for a node on the fault with tributary area A. Following the conventions
Fg1-q2 Tq Fr1-r2
=Ah
Tr (8.4)Fp1+P2 0
Fq1+q2 0
Fr-1+r2 0
The initial tractions do not contribute any forces to the average degrees of freedom, which we associate with movement of both sides of the fault in the same direction, because we assume that the fault surface is in equilibrium.
The friction force does not require any transformation; the product of the coefficient of friction and the force acting on the relative normal degree of freedom gives the magnitude of the friction force vector acting on the slip degree of freedom. The dynamic deformation in the domain may cause variations in the normal forces acting on the fault. We compute the dynamic normal force at the slip degrees of freedom as part of the formulation of the right-hand side of the time stepping equation (equation (8.3)). By checking that the normal force remains compressive, we confirm that clamping the relative normal displacement across the fault remains valid. In other words, because tensile tractions imply opening of the fault, which we do not allow, we want the normal tractions to remain compressive.
The appearance of the difference between the tectonic force vector and the friction force vector in the equation of motion implies that we may create the same sliding behavior from an infinite combination of tectonic and friction forces by keeping the difference between them the same. In other words, given the sliding behavior and the values of the tectonic forces and friction forces, if we are given different tectonic forces, we may adjust the friction model to maintain the same sliding behavior.
8.1.3 Initiation of Sliding
To initiate sliding at a point on the fault, the friction force must be less than the sum of the other forces acting on that point. We start the earthquakes by increasing the tectonic forces in order to overcome the friction. During the first 0.5 sec of the simulation, we increase the friction force above the level given by the friction model in order to create gradual failure and prevent sudden initiation of the rupture. At each point on the fault where failure occurs, we transition the friction force from
the critical force necessary to prevent slip at time t = 0 to the force given by the friction model at time t = 0.5 sec. Equation (8.5) gives the expression for the transition of the artificially adjusted friction force, FjdJ, from the critical force required to prevent slip, Fer, to the force given by the friction model.
Fadj f
8.1.4 Condition for Termination of ~:Hiding
(8.5)
Once a point on the fault starts sliding, it continues sliding until the slip rate decreases and the friction force is large enough to "lock" the fault. In more mathematical terms, the sliding stops when the friction force becomes larger than the sum of all of the other forces. Because sliding occurs on a plane, we must consider the vectors for the forces acting on each node on the fault .. Following the conventions of section 2.4.1, we consider the Pl - P2 and qi - q2 sliding degrees of freedom at each node on the fault. equation (8.3) with diagonal mass and damping matrices, the magnitude of the critical friction force for the ith degree of freedom is
Fcri (Di(t), Di(t At))
[Ft, - A~ 2 Mi
(ui(t +At) - 2u.;(t) + u;(t - At)) -2 ~t
Ci(ui(t +At) - ui(t - At)) - KijUj(t)I·
(8.6) We do not know the value of ui(t flt) since that is what we want to find; however, when sliding stops, ui(t +At) = ui(t), and the critical friction force for the ith degree of freedom becomes
Fer; (D(t), D(t At))=
IFti + A1
t2Mi (u;(t) ui(t At))
2 ~tci
(ui(t) ui(t At)) KiJui(t)/. (8.7) When sliding stops, the friction force may act in any direction. This implies we do not need to account for the current sliding direction in our criterion for terminating the sliding. In other words, we only need to determine if the magnitude of the friction force meets or exceeds the magnitude of the force required to terminate the sliding. Denoting the friction force at a node on the fault by F1, equation (8.8) gives the expression for the condition used to determine if sliding stops at a node on the fault with sliding degrees of freedom Pl - P2 and qi - q2·(8.8)
slip occurred and Ai is the fault tributary area for node i. We follow the same convention that we use for stress and denote the friction forces before and after by Fo and F1. A positive stress drop signifies a reduction in the stress on the fault.
(8.9)